JEE Main & Advanced Mathematics Definite Integration Question Bank

done Properties of Definite Integration Total Questions - 145

• question_answer1)
  \[\int_{0}^{\pi }{x}\,f\,(\sin x)\,dx=\]     [IIT 1982; Kurukshetra CEE 1993]
  A)
                  \[\pi \int_{0}^{\pi }{f(\sin x)\,dx}\]          done clear
  B)
                  \[\frac{\pi }{2}\int_{0}^{\pi }{f(\sin x)\,dx}\] done clear
  C)
                  \[\frac{\pi }{2}\int_{0}^{\pi /2}{f(\sin x)\,dx}\]   done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{-4}^{4}{|x+2|\,dx}=\]
  A)
                  50           done clear
  B)
                  24 done clear
  C)
                  20           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer3)
  \[\int_{0}^{\pi /2}{\frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}}\,dx=}\] [MP PET 1990, 95; IIT 1983; MNR 1990]
  A)
                  \[\pi \]  done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  \[\frac{\pi }{3}\] done clear
  View Solution play_arrow

• question_answer4)
  \[\int_{0}^{\pi /2}{\frac{d\theta }{1+\tan \theta }}=\] [Roorkee 1980; MP PET 1996; DCE 1999]
  A)
                  \[\pi \]  done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{\pi }{3}\]               done clear
  D)
                  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer5)
  If \[f(x)=\int_{a}^{x}{{{t}^{3}}{{e}^{t}}\,dt\,,}\] then \[\frac{d}{dx}\,f(x)=\]         [MP PET 1989]
  A)
                  \[{{e}^{x}}({{x}^{3}}+3{{x}^{2}})\]            done clear
  B)
                  \[{{x}^{3}}{{e}^{x}}\] done clear
  C)
                  \[{{a}^{3}}{{e}^{a}}\]      done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer6)
  \[\int_{-1}^{1}{x\,|x|\,}dx=\] [MP PET 1990; Pb. CET 2004]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  2              done clear
  D)
                  \[-2\] done clear
  View Solution play_arrow

• question_answer7)
  \[\int_{0}^{\pi }{x\log \sin x}\,dx=\]
  A)
                  \[\frac{\pi }{2}\log \frac{1}{2}\] done clear
  B)
                  \[\frac{{{\pi }^{2}}}{2}\log \frac{1}{2}\] done clear
  C)
                  \[\pi \log \frac{1}{2}\]    done clear
  D)
                  \[{{\pi }^{2}}\log \frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{0}^{\pi /2}{\,\,\log \tan x\,dx=}\]               [MP PET 1999; RPET 2001, 02; Karnataka CET 1999, 2000, 01, 02]
  A)
                  \[\frac{\pi }{2}{{\log }_{e}}2\]     done clear
  B)
                  \[-\frac{\pi }{2}{{\log }_{e}}2\] done clear
  C)
                  \[\pi {{\log }_{e}}2\]       done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{0}^{\pi /2}{{}}\log \sin x\,dx=\]        [MP PET 1994; RPET 1995, 96, 97]
  A)
                  \[-\left( \frac{\pi }{2} \right)\log 2\]         done clear
  B)
                  \[\pi \log \frac{1}{2}\] done clear
  C)
                  \[-\pi \log \frac{1}{2}\]  done clear
  D)
                  \[\frac{\pi }{2}\log 2\] done clear
  View Solution play_arrow

• question_answer10)
  \[\int_{0}^{\pi /2}{\frac{\cos x-\sin x}{1+\sin x\cos x}}\,dx=\]                                     [Karnataka CET 2004]
  A)
                  2              done clear
  B)
                  \[-2\] done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{-1}^{1}{\log \frac{2-x}{2+x}\,dx}=\]   [Roorkee 1986; Kurukshetra CEE 1998]
  A)
                  2              done clear
  B)
                  1 done clear
  C)
                  \[-1\]     done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer12)
  \[\int_{-1}^{1}{{{x}^{17}}{{\cos }^{4}}x}\,dx=\]                                      [MP PET 1990]
  A)
                  \[-2\]     done clear
  B)
                  \[-1\] done clear
  C)
                  0              done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{0}^{\pi /2}{\frac{{{\sin }^{3/2}}x\,dx}{{{\cos }^{3/2}}x+{{\sin }^{3/2}}x}}=\] [Roorkee 1989; BIT Ranchi 1989]
  A)
                  0              done clear
  B)
                  \[\pi \] done clear
  C)
                  \[\pi /2\]              done clear
  D)
                  \[\pi /4\] done clear
  View Solution play_arrow

• question_answer14)
  \[\int_{-\pi /2}^{\pi /2}{\sqrt{\frac{1}{2}(1-\cos 2x)}}\,dx=\]
  A)
                  0              done clear
  B)
                  2 done clear
  C)
                  \[\frac{1}{2}\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer15)
  \[\int_{-\pi /2}^{\pi /2}{\log \left( \frac{2-\sin \theta }{2+\sin \theta } \right)\,d\theta =}\]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  2              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer16)
  If \[f(x)=\left\{ \begin{matrix}    4x+3\,, & \text{if} & 1\le x\le 2  \\    3x+5\,, & \text{if} & 2<x\le 4  \\ \end{matrix} \right.\] then \[\int_{1}^{4}{\,f(x)}\,dx=\]
  A)
                  80           done clear
  B)
                  20 done clear
  C)
                  \[-20\]  done clear
  D)
                  37 done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{0}^{\pi /4}{\log (1+\tan \theta )\,d\theta =}\] [SCRA 1986; Karnataka CET 2000, 05]
  A)
                  \[\frac{\pi }{4}\log 2\]    done clear
  B)
                  \[\frac{\pi }{4}\log \frac{1}{2}\] done clear
  C)
                  \[\frac{\pi }{8}\log 2\]    done clear
  D)
                  \[\frac{\pi }{8}\log \frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{0}^{2\pi }{\frac{\sin 2\theta }{a-b\,\cos \theta }\,d\theta =}\]                                  [Roorkee 1988]
  A)
                  1              done clear
  B)
                  2 done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{0}^{1}{f(1-x)\,dx}\] has the same value as the integral [SCRA 1990]
  A)
                  \[\int_{0}^{1}{f(x)\,dx}\]              done clear
  B)
                  \[\int_{0}^{1}{f(-x)\,dx}\] done clear
  C)
                  \[\int_{0}^{1}{f(x-1)\,dx}\]          done clear
  D)
                  \[\int_{-1}^{1}{f(x)\,dx}\] done clear
  View Solution play_arrow

• question_answer20)
  The smallest interval \[[a,\,\,b]\] such that \[\int_{0}^{1}{\frac{dx}{\sqrt{1+{{x}^{4}}}}}\in [a,\,\,b]\] is given by
  A)
                  \[\left[ \frac{1}{\sqrt{2}},\,\,1 \right]\]  done clear
  B)
                  \[[0,\,\,1]\] done clear
  C)
                  \[\left[ \frac{1}{2},\,\,2 \right]\]                done clear
  D)
                  \[\left[ \frac{3}{4},\,\,1 \right]\] done clear
  View Solution play_arrow

• question_answer21)
  Assume that \[f\] is continuous everywhere, then \[\frac{1}{c}\int_{ac}^{bc}{f\left( \frac{x}{c} \right)}\,dx=\]
  A)
                  \[\int_{a}^{b}{f\left( \frac{x}{c} \right)}\,dx\]     done clear
  B)
                  \[\frac{1}{c}\int_{a}^{b}{f(x)\,dx}\] done clear
  C)
                  \[\int_{a}^{b}{f(x)\,dx}\]             done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer22)
  \[\int_{\,-1/2}^{\,1/2}{(\cos x)\,\left[ \log \left( \frac{1-x}{1+x} \right) \right]\,dx=}\]                    [Karnataka CET 2002]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[{{e}^{1/2}}\]   done clear
  D)
                  \[2{{e}^{1/2}}\] done clear
  View Solution play_arrow

• question_answer23)
  The value of \[\int_{\,0}^{\,1}{\,\frac{dx}{x+\sqrt{1-{{x}^{2}}}}}\] is                        [MP PET 2003]
  A)
                  \[\frac{\pi }{3}\]               done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{1}{2}\]    done clear
  D)
                  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer24)
  If \[\int_{-1}^{1}{f(x)\,dx=0}\], then                                       [SCRA 1990]
  A)
                  \[f(x)=f(-x)\]      done clear
  B)
                  \[f(-x)=-f(x)\] done clear
  C)
                  \[f(x)=2f(x)\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer25)
  \[\int_{-1}^{1}{|1-x|dx}=\]                                         [Karnataka CET 2004]
  A)
                  ? 2          done clear
  B)
                  0 done clear
  C)
                  2              done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer26)
  If n is a positive integer and [x] is the greatest integer not exceeding x, then \[\int_{0}^{n}{\,\,\{x-[x]\}\,dx}\] equals
  A)
                  \[{{n}^{2}}/2\]   done clear
  B)
                  \[n(n-1)/2\] done clear
  C)
                  \[n\,/\,2\]           done clear
  D)
                  \[\frac{{{n}^{2}}}{2}-n\] done clear
  View Solution play_arrow

• question_answer27)
  \[\int_{0}^{\pi }{x{{\sin }^{3}}x\,dx}=\]                                  [CEE 1993]
  A)
                  \[\frac{4\pi }{3}\]             done clear
  B)
                  \[\frac{2\pi }{3}\] done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer28)
  \[\int_{-2}^{2}{|1-{{x}^{2}}|\,dx=}\] [IIT 1989; BIT Mesra 1996; Kurukshetra CEE 1998;  MP PET 2002; Kerala (Engg.) 2002]
  A)
                  2              done clear
  B)
                  4 done clear
  C)
                  6              done clear
  D)
                  8 done clear
  View Solution play_arrow

• question_answer29)
  \[\int_{0}^{\pi /2}{\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx=}\]             [MNR 1989; UPSEAT 2002]
  A)
                  0              done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{0}^{\pi /2}{\frac{x\sin x\cos x}{{{\cos }^{4}}x+{{\sin }^{4}}x}}\,dx=\]                                        [IIT 1985]
  A)
                  0              done clear
  B)
                  \[\frac{\pi }{8}\] done clear
  C)
                  \[\frac{{{\pi }^{2}}}{8}\] done clear
  D)
                  \[\frac{{{\pi }^{2}}}{16}\] done clear
  View Solution play_arrow

• question_answer31)
  The correct evaluation of \[\int_{0}^{\pi /2}{\left| \,\sin \left( x-\frac{\pi }{4} \right)\, \right|\,dx}\] is [MP PET 1993]
  A)
                  \[2+\sqrt{2}\]    done clear
  B)
                  \[2-\sqrt{2}\] done clear
  C)
                  \[-2+\sqrt{2}\]  done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer32)
  \[\int_{0}^{a}{f(x)\,dx}=\]                                       [BIT Ranchi 1992]
  A)
                  \[\int_{0}^{a}{f(a+x)\,dx}\]         done clear
  B)
                  \[\int_{0}^{a}{f(2a+x)\,dx}\] done clear
  C)
                  \[\int_{0}^{a}{f(x-a)\,dx}\]          done clear
  D)
                  \[\int_{0}^{a}{f(a-x)\,dx}\] done clear
  View Solution play_arrow

• question_answer33)
  \[\int_{0}^{\pi /2}{\,\,\,\,\,|\sin x-\cos x|\,dx=}\] [Roorkee 1990; MP PET 2001; UPSEAT 2001]
  A)
                  0              done clear
  B)
                  \[2(\sqrt{2}-1)\] done clear
  C)
                  \[\sqrt{2}-1\]     done clear
  D)
                  \[2(\sqrt{2}+1)\] done clear
  View Solution play_arrow

• question_answer34)
  \[\int_{0}^{\pi }{|\cos x|\,dx=}\]           [MP PET 1998; Pb. CET 2001]
  A)
                  \[\pi \]  done clear
  B)
                  0 done clear
  C)
                  2              done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer35)
  The value of the integral \[\int_{-\pi /4}^{\pi /4}{{{\sin }^{-4}}x}\,dx\] is [IIT Screening; MP PET 2003]
  A)
                  3/2         done clear
  B)
                  ?8/3 done clear
  C)
                  3/8         done clear
  D)
                  8/3 done clear
  View Solution play_arrow

• question_answer36)
  \[\int_{0}^{1.5}{[{{x}^{2}}]\,dx}\], where \[[\,\,.\,\,]\]denotes the greatest integer function, equals           [IIT 1988; DCE 2000, 01]
  A)
                  \[2+\sqrt{2}\]    done clear
  B)
                  \[2-\sqrt{2}\] done clear
  C)
                  \[-2+\sqrt{2}\]  done clear
  D)
                  \[-2-\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{0}^{\pi }{\frac{x\tan x}{\sec x+\tan x}}\,dx=\]                                     [MNR 1984]
  A)
                  \[\frac{\pi }{2}-1\]           done clear
  B)
                  \[\pi \left( \frac{\pi }{2}+1 \right)\] done clear
  C)
                  \[\frac{\pi }{2}+1\]          done clear
  D)
                  \[\pi \left( \frac{\pi }{2}-1 \right)\] done clear
  View Solution play_arrow

• question_answer38)
  \[\int_{0}^{\pi }{\frac{x\,\tan x}{\sec x+\cos x}}\,dx=\] [MNR 1985; BIT Ranchi 1986; UPSEAT 2002]
  A)
                  \[\frac{{{\pi }^{2}}}{4}\] done clear
  B)
                  \[\frac{{{\pi }^{2}}}{2}\] done clear
  C)
                  \[\frac{3{{\pi }^{2}}}{2}\]              done clear
  D)
                  \[\frac{{{\pi }^{2}}}{3}\] done clear
  View Solution play_arrow

• question_answer39)
  \[\int_{-1}^{1}{{{\sin }^{3}}x{{\cos }^{2}}x\,dx=}\]            [MNR 1991; UPSEAT 2000]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[\frac{1}{2}\]    done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer40)
  For any integer \[n,\] the integral                 \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\]                                            [MNR 1982]
  A)
                  \[-1\]     done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer41)
  \[\int_{1/e}^{e}{|\log x|\,dx=}\]                                             [UPSEAT 2001]
  A)
                  \[1-\frac{1}{e}\]               done clear
  B)
                  \[2\,\left( 1-\frac{1}{e} \right)\] done clear
  C)
                  \[{{e}^{-1}}-1\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer42)
  \[\int_{\,0}^{\,\pi /2}{\{x-[\sin x]\}\,dx}\] is equal to [AMU 1999]
  A)
                  \[\frac{{{\pi }^{2}}}{8}\] done clear
  B)
                  \[\frac{{{\pi }^{2}}}{8}-1\] done clear
  C)
                  \[\frac{{{\pi }^{2}}}{8}-2\]            done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer43)
  The value of the integral \[I=\int_{\,0}^{\,1}{\,x{{(1-x)}^{n}}dx}\] is [AIEEE 2003]
  A)
                  \[\frac{1}{n+1}\]              done clear
  B)
                  \[\frac{1}{n+2}\] done clear
  C)
                  \[\frac{1}{n+1}-\frac{1}{n+2}\]  done clear
  D)
                  \[\frac{1}{n+1}+\frac{1}{n+2}\] done clear
  View Solution play_arrow

• question_answer44)
  The value of \[\int_{\pi }^{2\pi }{[2\sin x]\,dx,}\] where \[[\,\,.\,\,]\] represents the greatest integer function, is                                         [IIT 1995]
  A)
                  \[-\pi \] done clear
  B)
                  \[-2\pi \] done clear
  C)
                  \[-\frac{5\pi }{3}\]           done clear
  D)
                  \[\frac{5\pi }{3}\] done clear
  View Solution play_arrow

• question_answer45)
  If \[f(x)\] is a continuous periodic function with period \[T,\] then the integral \[I=\int_{a}^{a+T}{f(x)\,dx}\] is
  A)
                  Equal to \[2a\]   done clear
  B)
                  Equal to \[3a\] done clear
  C)
                  Independent of \[a\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer46)
  If \[\int_{0}^{\pi }{x\,f({{\cos }^{2}}x+{{\tan }^{4}}x)\,dx}\] \[=k\int_{0}^{\pi /2}{f({{\cos }^{2}}x+{{\tan }^{4}}x)\,dx,}\] then the value of \[k\] is
  A)
                  \[\frac{\pi }{2}\]               done clear
  B)
                  \[\pi \] done clear
  C)
                  \[-\frac{\pi }{2}\]              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer47)
  \[\int_{-3}^{3}{\frac{{{x}^{2}}\sin x}{1+{{x}^{6}}}\,dx=}\]
  A)
                  4              done clear
  B)
                  2 done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer48)
  The value of \[\int_{0}^{\pi /2}{\frac{dx}{1+{{\tan }^{3}}x}}\] is    [IIT 1993; DCE 2000, 01]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[\frac{\pi }{2}\]               done clear
  D)
                  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer49)
  The value of \[\int_{\pi /4}^{3\pi /4}{\frac{\varphi }{1+\sin \varphi }\,d\varphi ,}\] is [AI CBSE 1990; IIT 1993]
  A)
                  \[\pi \tan \frac{\pi }{8}\]               done clear
  B)
                  \[\log \tan \frac{\pi }{8}\] done clear
  C)
                  \[\tan \frac{\pi }{8}\]      done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer50)
  If \[f(a+b-x)=f(x),\] then \[\int_{a}^{b}{x\,f(x)\,dx=}\] [CEE 1993; AIEEE 2003]
  A)
                  \[\frac{a+b}{2}\int_{a}^{b}{f(b-x)\,dx}\]               done clear
  B)
                  \[\frac{a+b}{2}\int_{a}^{b}{f(x)\,dx}\] done clear
  C)
                  \[\frac{b-a}{2}\int_{a}^{b}{f(x)\,dx}\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer51)
  \[\int_{0}^{\pi }{x\sin x\,dx=}\]                                 [SCRA 1980, 91]
  A)
                  \[\pi \]  done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  \[{{\pi }^{2}}\] done clear
  View Solution play_arrow

• question_answer52)
  If \[\int_{-a}^{a}{\sqrt{\frac{a-x}{a+x}}\,dx=k\pi ,}\] then \[k=\]  [AISSE 1986; SCRA 1986]
  A)
                  \[-a\]     done clear
  B)
                  \[-2a\] done clear
  C)
                  \[2a\]    done clear
  D)
                  \[a\] done clear
  View Solution play_arrow

• question_answer53)
  If \[\int_{0}^{2a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}}\] then                           [SCRA 1986]
  A)
                  \[f(2a-x)=-f(x)\]               done clear
  B)
                  \[f(2a-x)=f(x)\] done clear
  C)
                  \[f(a-x)=-f(x)\]  done clear
  D)
                  \[f(a-x)=f(x)\] done clear
  View Solution play_arrow

• question_answer54)
  If \[I=\int_{0}^{\pi /4}{\,{{\sin }^{2}}x\,dx}\]and\[J=\int_{0}^{\pi /4}{{{\cos }^{2}}x\,dx,}\] then \[I=\]  [SCRA 1989]
  A)
                  \[\frac{\pi }{4}-J\]            done clear
  B)
                  \[2J\] done clear
  C)
                  \[J\]       done clear
  D)
                  \[\frac{J}{2}\] done clear
  View Solution play_arrow

• question_answer55)
  The value of \[\int_{1}^{5}{(|x-3|+|1-x|)\,dx}\] is           [IIT Screening]
  A)
                  10           done clear
  B)
                  \[\frac{5}{6}\] done clear
  C)
                  21           done clear
  D)
                  12 done clear
  View Solution play_arrow

• question_answer56)
  The value of \[\int_{2}^{3}{\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}}\,dx\] is [IIT 1994; Kurukshetra CEE 1998]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  \[-1\] done clear
  D)
                  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer57)
  The value of \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3x}\,dx\] is           [Bihar CEE 1994]
  A)
                  1              done clear
  B)
                  \[-1\] done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer58)
  \[\int_{0}^{\pi /2}{\frac{1}{1+\sqrt{\tan x}}}\,dx=\]  [RPET 1995; Kurukshetra CEE 1998]
  A)
                  \[\frac{\pi }{2}\]               done clear
  B)
                  \[\frac{\pi }{4}\] done clear
  C)
                  \[\frac{\pi }{6}\]               done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer59)
  The value of \[\int_{-1}^{1}{\frac{\sin x-{{x}^{2}}}{3-|x|}\,dx}\] is                            [Roorkee 1995]
  A)
                  0              done clear
  B)
                  \[2\int_{0}^{1}{\frac{\sin x}{3-|x|}\,dx}\] done clear
  C)
                  \[2\int_{0}^{1}{\frac{-{{x}^{2}}}{3-|x|}}\,dx\]    done clear
  D)
                  \[2\int_{0}^{1}{\frac{\sin x-{{x}^{2}}}{3-|x|}\,dx}\] done clear
  View Solution play_arrow

• question_answer60)
  \[\int_{-1}^{1}{{{\sin }^{11}}x\,dx}\] is equal to                                 [MNR 1995]
  A)
                  \[\frac{10}{11}.\frac{8}{9}.\frac{6}{7}.\frac{4}{5}.\frac{2}{3}\]      done clear
  B)
                  \[\frac{10}{11}.\frac{8}{9}.\frac{6}{7}.\frac{4}{5}.\frac{2}{3}.\frac{\pi }{2}\] done clear
  C)
                  1              done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer61)
  If \[(n-m)\] is odd and \[|m|\,\ne \,|n|,\] then \[\int_{0}^{\pi }{\cos mx\sin nx}\,dx\] is
  A)
                  \[\frac{2n}{{{n}^{2}}-{{m}^{2}}}\]             done clear
  B)
                  0 done clear
  C)
                  \[\frac{2n}{{{m}^{2}}-{{n}^{2}}}\]             done clear
  D)
                  \[\frac{2m}{{{n}^{2}}-{{m}^{2}}}\] done clear
  View Solution play_arrow

• question_answer62)
  To find the numerical value of \[\int_{-2}^{2}{(p{{x}^{2}}+qx+s)\,dx,}\] it is necessary to know the values of constants                [IIT 1992]
  A)
                  \[p\]      done clear
  B)
                  \[q\] done clear
  C)
                  \[s\]       done clear
  D)
                  \[p\] and \[s\] done clear
  View Solution play_arrow

• question_answer63)
  If \[I=\int_{0}^{100\pi }{\sqrt{(1-\cos 2x)}\,dx,}\]then the value of \[I\] is
  A)
                  \[100\sqrt{2}\]  done clear
  B)
                  \[200\sqrt{2}\] done clear
  C)
                  \[50\sqrt{2}\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer64)
  \[\int_{-1}^{1}{\log \left( \frac{1+x}{1-x} \right)\,dx=}\]                [MP PET 1995; Pb. CET 2000]
  A)
                  2              done clear
  B)
                  1 done clear
  C)
                  0              done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer65)
  If \[\int_{-1}^{4}{f(x)\,dx}=4\] and \[\int_{2}^{4}{(3-f(x))\,dx=7,}\] then the value of \[\int_{2}^{-1}{f(x)\,dx}\]  is
  A)
                  2              done clear
  B)
                  ? 3 done clear
  C)
                  ? 5          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer66)
  The function \[F(x)=\int_{0}^{x}{\log \left( \frac{1-x}{1+x} \right)}\,dx\] is
  A)
                  An even function             done clear
  B)
                  An odd function done clear
  C)
                  A periodic function          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\int_{-\pi /2}^{\pi /2}{\frac{\cos x}{1+{{e}^{x}}}\,dx=}\]                                            [EAMCET 1992]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  \[-1\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer68)
  The value of \[\int_{-1}^{1}{(\sqrt{1+x+{{x}^{2}}}-\sqrt{1-x+{{x}^{2}}})\,dx}\] is
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[-1\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer69)
  The value of \[\int_{0}^{\pi /2}{\log \,\left( \frac{4+3\sin x}{4+3\cos x} \right)}\,dx\] is
  A)
                  2              done clear
  B)
                  \[\frac{3}{4}\] done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer70)
  The value of \[\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{2x-1}{1+x-{{x}^{2}}} \right)}\,dx\] is
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  \[-1\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer71)
  The value of \[\int_{0}^{2\pi }{{{\cos }^{99}}x\,dx}\] is
  A)
                  1              done clear
  B)
                  \[-1\] done clear
  C)
                  99           done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer72)
  If \[[x]\] denotes the greatest integer less than or equal to \[x,\] then the value of the integral \[\int_{0}^{2}{{{x}^{2}}[x]\,dx}\] equals [Kurukshetra CEE 1996; Pb. CET 2001]
  A)
                  5/3         done clear
  B)
                  7/3 done clear
  C)
                  8/3         done clear
  D)
                  4/3 done clear
  View Solution play_arrow

• question_answer73)
  \[\int_{\,0}^{\,\pi }{{{\cos }^{3}}x\,dx=}\]            [MP PET 1996; Pb. CET 2002]
  A)
                  \[-1\]     done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer74)
  \[\int_{\,0}^{\,2\pi }{|\sin x|\,dx=}\]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  2              done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer75)
  \[\int_{-3}^{3}{\frac{{{x}^{2}}\sin 2x}{{{x}^{2}}+1}\,dx=}\]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  \[2{{\log }_{e}}3\]            done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer76)
  \[\int_{\,0}^{\,\pi }{\log {{\sin }^{2}}x\,dx=}\]                                    [MP PET 1997]
  A)
                  \[2\pi {{\log }_{e}}\left( \frac{1}{2} \right)\]         done clear
  B)
                  \[\pi {{\log }_{e}}2+c\] done clear
  C)
                  \[\frac{\pi }{2}{{\log }_{e}}\left( \frac{1}{2} \right)+c\]    done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer77)
  If \[f(x)\] is an odd function of \[x,\] then \[\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{f(\cos x)\,dx}\] is equal to                                                          [MP PET 1998]
  A)
                  0              done clear
  B)
                  \[\int_{0}^{\frac{\pi }{2}}{f(\cos x)\,dx}\] done clear
  C)
                  \[2\int_{0}^{\frac{\pi }{2}}{f(\sin x)\,dx}\]            done clear
  D)
                  \[\int_{0}^{\pi }{f(\cos x)\,dx}\] done clear
  View Solution play_arrow

• question_answer78)
  \[\int_{0}^{\pi }{{{\sin }^{2}}x\,dx}\] is equal to                                 [MP PET 1999]
  A)
                  \[\pi \]  done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer79)
  \[\int_{0}^{\pi /2}{\frac{\sin x}{\sin x+\cos x}\,dx}\] equals [RPET 1996; Kerala (Engg.) 2002]
  A)
                  \[\frac{\pi }{2}\]               done clear
  B)
                  \[\frac{\pi }{3}\] done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  \[\frac{\pi }{6}\] done clear
  View Solution play_arrow

• question_answer80)
  \[\int_{-1}^{1}{x{{\tan }^{-1}}x\,dx}\] equals                                      [RPET 1997]
  A)
                  \[\left( \frac{\pi }{2}-1 \right)\]  done clear
  B)
                  \[\left( \frac{\pi }{2}+1 \right)\] done clear
  C)
                  \[(\pi -1)\]           done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer81)
  \[\int_{-a}^{a}{\sin x\,f(\cos x)\,dx=}\]                                 [RPET 1997]
  A)
                  \[2\int_{0}^{a}{\sin x\,f(\cos x)\,dx}\]   done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer82)
  The value of \[\int_{0}^{2\pi }{|{{\sin }^{3}}\theta |\,d\theta }\] is    [Roorkee Qualifying 1998]
  A)
                  0              done clear
  B)
                  \[3/8\] done clear
  C)
                  \[8/3\]  done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer83)
  \[\int_{\,-1}^{\,2}{|x|\,dx}\]                                      [DCE 1999]
  A)
                  5/2         done clear
  B)
                  1/2 done clear
  C)
                  3/2         done clear
  D)
                  7/2 done clear
  View Solution play_arrow

• question_answer84)
  \[\int_{\,0}^{\,3}{|2-x|dx}\] equals                                        [RPET 1999]
  A)
                  2/7         done clear
  B)
                  5/2 done clear
  C)
                  3/2         done clear
  D)
                  \[-3/2\] done clear
  View Solution play_arrow

• question_answer85)
  The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{2}^{\sin x}}}{{{2}^{\sin x}}+{{2}^{\cos x}}}dx}\] is [Karnataka CET 1999; Kerala (Engg.) 2005]
  A)
                  \[\frac{\pi }{4}\]               done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\pi \]  done clear
  D)
                  \[2\pi \] done clear
  View Solution play_arrow

• question_answer86)
  The value of \[\int_{\,0}^{\,1}{\,|\,3{{x}^{2}}-1\,|\,dx}\] is                          [AMU 1999]
  A)
                  0              done clear
  B)
                  \[4/3\sqrt{3}\] done clear
  C)
                  3/7         done clear
  D)
                  5/6 done clear
  View Solution play_arrow

• question_answer87)
  \[\int_{-\,\pi /2}^{\,\pi /2}{\,\frac{\sin x}{1+{{\cos }^{2}}x}{{e}^{-{{\cos }^{2}}x}}dx}\] is equal to              [AMU 1999]
  A)
                  \[2{{e}^{-1}}\]   done clear
  B)
                  1 done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer88)
  \[f(x)=f(2-x),\] then \[\int_{\,0.5}^{\,1.5}{\,xf(x)\,dx}\] equals [AMU 1999]
  A)
                  \[\int_{\,0}^{\,1}{\,f(x)\,dx}\]    done clear
  B)
                  \[\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\] done clear
  C)
                  \[2\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]          done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer89)
  The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{e}^{{{x}^{2}}}}}{{{e}^{{{x}^{2}}}}+{{e}^{{{\left( \frac{\pi }{2}\,\,-\,\,x \right)}^{2}}}}}dx}\] is    [AMU 1999]
  A)
                  \[\pi /4\]              done clear
  B)
                  \[\pi /2\] done clear
  C)
                  \[{{e}^{{{\pi }^{2}}/16}}\]             done clear
  D)
                  \[{{e}^{{{\pi }^{2}}/4}}\] done clear
  View Solution play_arrow

• question_answer90)
  If [x] denotes the greatest integer less than or equal to x, then the value of \[\int_{\,1}^{\,5}{\,\,[|x-3|]\,dx}\] is                 [Roorkee 1999]
  A)
                  1              done clear
  B)
                  2 done clear
  C)
                  4              done clear
  D)
                  8 done clear
  View Solution play_arrow

• question_answer91)
  \[\int_{\,-2}^{\,2}{|x|\,dx=}\]                                   [MP PET 2000]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  2              done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer92)
  Suppose f is such that \[f(-x)=-f(x)\] for every real x and \[\int_{\,0}^{\,1}{f(x)\,dx=5,}\] then \[\int_{\,-\,1}^{\,0}{f(t)\,dt=}\]                     [MP PET 2000]
  A)
                  10           done clear
  B)
                  5 done clear
  C)
                  0              done clear
  D)
                  ? 5 done clear
  View Solution play_arrow

• question_answer93)
  Let \[{{I}_{1}}=\int_{a}^{\pi -a}{xf(\sin x)dx,\,{{I}_{2}}=\int_{a}^{\pi -a}{\,\,f(\sin x)dx}}\], then \[{{I}_{2}}\] is equal to                                                            [AMU 2000]
  A)
                  \[\frac{\pi }{2}{{I}_{1}}\]               done clear
  B)
                  \[\pi \,{{I}_{1}}\] done clear
  C)
                  \[\frac{2}{\pi }{{I}_{1}}\]               done clear
  D)
                  \[2{{I}_{1}}\] done clear
  View Solution play_arrow

• question_answer94)
  \[\int_{-\frac{1}{2}}^{\,\frac{1}{2}}{\cos x\,\ln \frac{1+x}{1-x}dx}\] is equal to                     [AMU 2000]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  2              done clear
  D)
                  ln 3 done clear
  View Solution play_arrow

• question_answer95)
  The value of \[\int_{\,{{e}^{-1}}}^{\,{{e}^{2}}}{\left| \frac{{{\log }_{e}}x}{x} \right|\,dx}\] is                        [IIT Screening 2000]
  A)
                  \[\frac{3}{2}\]    done clear
  B)
                  \[\frac{5}{2}\] done clear
  C)
                  3              done clear
  D)
                  5 done clear
  View Solution play_arrow

• question_answer96)
  If \[f(x)=\left\{ \begin{matrix}    {{e}^{\cos x}}\sin x, & |x|\,\le 2  \\    2, & \text{otherwise}  \\ \end{matrix} \right.\], then \[\int_{\,-\,2}^{\,3}{f(x)\,dx}\] is equal to                                                    [IIT Screening 2000]
  A)
                  0              done clear
  B)
                  1 done clear
  C)
                  2              done clear
  D)
                  3 done clear
  View Solution play_arrow

• question_answer97)
  If \[f:R\to R\] and \[g:R\to R\] are one to one, real valued functions, then the value of the integral \[\int_{\,-\pi }^{\,\pi }{[f(x)+f(-x)]\,[g(x)-g(-x)]\,dx}\] is                 [DCE 2001; MP PET 2004]
  A)
                  0              done clear
  B)
                  \[\frac{8}{3}\] done clear
  C)
                  1              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer98)
  \[\int_{\,\pi /6}^{\,\pi /3}{\,\frac{dx}{1+\sqrt{\cot x}}}\] is                                           [DCE 2001]
  A)
                  \[\pi /3\]              done clear
  B)
                  \[\pi /6\] done clear
  C)
                  \[\pi /12\] done clear
  D)
                  \[\pi /2\] done clear
  View Solution play_arrow

• question_answer99)
  The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{\sin }^{2/3}}x}{{{\sin }^{2/3}}x+{{\cos }^{2/3}}x}dx}\] is            [RPET 2001]
  A)
                  \[\pi /4\]              done clear
  B)
                  \[\pi /2\] done clear
  C)
                  \[3\pi /4\]           done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer100)
  \[\int_{\,-\,1}^{\,1}{\log (x+\sqrt{{{x}^{2}}+1})\,dx=}\]                                  [MP PET 2001]
  A)
                  0              done clear
  B)
                  log 2 done clear
  C)
                  \[\log \frac{1}{2}\]           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer101)
  The value of the integral \[\int_{-\pi }^{\pi }{{{(\cos ax-\sin bx)}^{2}}dx}\], (a and b are integer) is                                            [UPSEAT 2001]
  A)
                  \[-\pi \] done clear
  B)
                  0 done clear
  C)
                  \[\pi \]  done clear
  D)
                  \[2\pi \] done clear
  View Solution play_arrow

• question_answer102)
  \[\int_{\,0}^{\,\pi }{\sqrt{\frac{1+\cos 2x}{2}}\,dx}\]  is equal to                                [AMU 2001]
  A)
                  0              done clear
  B)
                  2 done clear
  C)
                  1              done clear
  D)
                  \[-1\] done clear
  View Solution play_arrow

• question_answer103)
  \[\int_{\,0}^{\,2a}{f(x)dx=}\]                                     [RPET 2002]
  A)
                  \[2\int_{\,0}^{\,a}{\,f(x)dx}\] done clear
  B)
                  0 done clear
  C)
                  \[\int_{\,0}^{\,a}{\,f(x)dx+\int_{\,0}^{\,a}{\,f(2a-x)dx}}\] done clear
  D)
                  \[\int_{\,0}^{\,a}{f(x)dx+}\int_{\,0}^{\,2a}{\,f(2a-x)dx}\] done clear
  View Solution play_arrow

• question_answer104)
  \[\int_{\,0}^{\,\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}x\,dx}\] is equals to                              [MP PET 2002]
  A)
                  \[-1\]     done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer105)
  Find the value of \[\int_{\,0}^{\,9}{[\sqrt{x}+2]dx},\] where [.] is the greatest integer function                                  [UPSEAT 2002]
  A)
                  31           done clear
  B)
                  22 done clear
  C)
                  23           done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer106)
  The value of \[\int_{\,0}^{\,\sqrt{2}}{[{{x}^{2}}]\,dx},\] where [.] is the greatest integer function                                             [AIEEE 2002]
  A)
                  \[2-\sqrt{2}\]     done clear
  B)
                  \[2+\sqrt{2}\] done clear
  C)
                  \[\sqrt{2}-1\]     done clear
  D)
                  \[\sqrt{2}-2\] done clear
  View Solution play_arrow

• question_answer107)
  \[\int_{\,0}^{\,1000}{{{e}^{x-[x]}}dx}\] is                                              [AMU 2002]
  A)
                  \[{{e}^{1000}}-1\] done clear
  B)
                  \[\frac{{{e}^{1000}}-1}{e-1}\] done clear
  C)
                  \[1000(e-1)\]     done clear
  D)
                  \[\frac{e-1}{1000}\] done clear
  View Solution play_arrow

• question_answer108)
  The value of the integral \[\int_{\,\frac{1}{n}}^{\,\frac{an-1}{n}}{\frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}}dx}\] is [AMU 2002]
  A)
                  \[\frac{a}{2}\]    done clear
  B)
                  \[\frac{na+2}{2n}\] done clear
  C)
                  \[\frac{na-2}{2n}\]          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer109)
  \[\int_{\,0}^{\,\pi /2}{\sin 2x\log \tan x\,dx}\] is equal to [Kerala (Engg.) 2002; AI CBSE 1990; Karnataka CET 1996, 98]
  A)
                  \[\pi \]  done clear
  B)
                  \[\pi /2\] done clear
  C)
                  0              done clear
  D)
                  \[2\pi \] done clear
  View Solution play_arrow

• question_answer110)
  The integral \[\int_{\,-1/2}^{\,1/2}{\,\left\{ [x]+\log \left( \frac{1+x}{1-x} \right) \right\}}\,dx\] equal (where [.] is the greatest integer function)                                            [IIT Screening 2002]
  A)
                  \[-\frac{1}{2}\]  done clear
  B)
                  0 done clear
  C)
                  1              done clear
  D)
                  \[2\log \frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer111)
  \[\int_{\,0}^{\,2\pi }{(\sin x+|\sin x|)\,dx=}\]                                    [Karnataka CET 2003]
  A)
                  0              done clear
  B)
                  4 done clear
  C)
                  8              done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer112)
  The value of \[\int_{-\pi /2}^{\,\pi /2}{(3\sin x+{{\sin }^{3}}x)\,dx}\] is     [MP PET 2003]
  A)
                  3              done clear
  B)
                  2 done clear
  C)
                  0              done clear
  D)
                  \[\frac{10}{3}\] done clear
  View Solution play_arrow

• question_answer113)
  The value of \[I=\int_{\,0}^{\,1}{\,x\,\left| x-\frac{1}{2} \right|\,dx}\] is                               [UPSEAT 2003]
  A)
                  1/3         done clear
  B)
                  1/4 done clear
  C)
                  1/8         done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer114)
  The value of \[\int_{\,0}^{\,8}{\,|x-5|\,dx}\] is                                 [UPSEAT 2003]
  A)
                  17           done clear
  B)
                  12 done clear
  C)
                  9              done clear
  D)
                  18 done clear
  View Solution play_arrow

• question_answer115)
  \[\int_{\,0}^{\,2}{\,|x-1|\,dx=}\]   [SCRA 1990; RPET 2001; UPSEAT 2003]
  A)
                  0 done clear
  B)
                  2 done clear
  C)
                  1/2         done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer116)
  \[\int_{\,-\,2}^{\,2}{\,\left| \,[x]\, \right|\,dx=}\]                                            [EAMCET 2003]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  3              done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer117)
  \[\int_{\,0}^{\,1}{\,{{\tan }^{-1}}\left( \frac{1}{{{x}^{2}}-x+1} \right)\,dx}\] is                                      [Orissa JEE 2003]
  A)
                  ln 2         done clear
  B)
                  \[-\ln 2\] done clear
  C)
                  \[\frac{\pi }{2}+\ln 2\]    done clear
  D)
                  \[\frac{\pi }{2}-\ln 2\] done clear
  View Solution play_arrow

• question_answer118)
  The value of \[\int_{\,a}^{\,b}{\frac{x}{|x|}dx,\,\,a<b<0}\] is     [Orissa JEE 2003]
  A)
                  \[-(|\,a\,|+|\,b\,|)\]     done clear
  B)
                  \[|b|-|a|\] done clear
  C)
                  \[|a|-|b|\]         done clear
  D)
                  \[|a|+|b|\] done clear
  View Solution play_arrow

• question_answer119)
  The value of \[\int_{\,-2}^{\,2}{\left[ p\ln \left( \frac{1+x}{1-x} \right)+q\ln {{\left( \frac{1-x}{1+x} \right)}^{-2}}+r \right]\,dx}\] depends on                                            [Orissa JEE 2003]
  A)
                  The value of p   done clear
  B)
                  The value of q done clear
  C)
                  The value of r    done clear
  D)
                  The value of p and q done clear
  View Solution play_arrow

• question_answer120)
  \[\int_{0}^{\pi }{\frac{xdx}{1+\sin x}}\]is equal to                                            [UPSEAT 2004]
  A)
                  \[-\pi \] done clear
  B)
                  \[\frac{\pi }{2}\] done clear
  C)
                  \[\pi \]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer121)
  The value of \[\int_{-2}^{3}{|1-{{x}^{2}}|dx}\]is                                [AIEEE 2004]
  A)
                  \[\frac{1}{3}\]    done clear
  B)
                  \[\frac{14}{3}\] done clear
  C)
                  \[\frac{7}{3}\]    done clear
  D)
                  \[\frac{28}{3}\] done clear
  View Solution play_arrow

• question_answer122)
  If \[f(x)=|x-1|\], then \[\int_{0}^{2}{f(x)dx}\]is                 [Orissa JEE 2004]
  A)
                  1              done clear
  B)
                  0 done clear
  C)
                  2              done clear
  D)
                  ?2 done clear
  View Solution play_arrow

• question_answer123)
  If \[\int_{0}^{\pi }{xf(\sin x)dx=A}\int_{0}^{\pi /2}{f(\sin x)dx}\], then A is           [AIEEE 2004]
  A)
                  \[2\pi \]                done clear
  B)
                  \[\pi \] done clear
  C)
                  \[\frac{\pi }{4}\]               done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer124)
  \[\int_{0}^{\pi /2}{{}}(\sin x-\cos x)\log (\sin x+\cos x)\,dx=\]    [SCRA 1986]
  A)
                  \[-1\]     done clear
  B)
                  1 done clear
  C)
                  0              done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer125)
  The function \[L(x)=\int_{1}^{x}{\frac{dt}{t}}\] satisfies the equation                 [IIT 1996; DCE 2001]
  A)
                  \[L(x+y)=L(x)+L(y)\]       done clear
  B)
                  \[L\left( \frac{x}{y} \right)=L(x)+L(y)\] done clear
  C)
                  \[L(xy)=L(x)+L(y)\]          done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer126)
  The value of integral \[\int_{0}^{1}{{{e}^{{{x}^{2}}}}}dx\] lies in interval                 [CEE 1993]
  A)
                  \[(0,\,\,1)\]         done clear
  B)
                  (\[-1,\,\,0)\] done clear
  C)
                  \[(1,\,\,e)\]        done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer127)
  If \[P=\int_{0}^{3\pi }{f({{\cos }^{2}}x)dx}\,\,\text{and}\,\,Q=\int_{0}^{\pi }{f({{\cos }^{2}}x)dx}\], then                                                 [Orissa JEE 2004]
  A)
                  \[P-Q=0\]            done clear
  B)
                  \[P-2Q=0\] done clear
  C)
                  \[P-3Q=0\]          done clear
  D)
                  \[P-5Q=0\] done clear
  View Solution play_arrow

• question_answer128)
  Let \[a,\,\,b,\,\,c\] be non-zero real numbers such that \[\int_{0}^{3}{(3a{{x}^{2}}+2bx+c)\,dx}=\int_{1}^{3}{(3a{{x}^{2}}+2bx+c})\,dx\,,\] then [BIT Ranchi 1991]
  A)
                  \[a+b+c=3\]       done clear
  B)
                  \[a+b+c=1\] done clear
  C)
                  \[a+b+c=0\]       done clear
  D)
                  \[a+b+c=2\] done clear
  View Solution play_arrow

• question_answer129)
  \[\int_{-\pi }^{\pi }{{{(\cos px-\sin qx)}^{2}}dx}\] is equal to (where \[p\] and \[q\] are integers) [IIT 1992]
  A)
                  \[-\pi \] done clear
  B)
                  0 done clear
  C)
                  \[\pi \]  done clear
  D)
                  \[2\pi \] done clear
  View Solution play_arrow

• question_answer130)
  If \[g(x)=\int_{0}^{x}{{{\cos }^{4}}t\,dt,}\] then \[g(x+\pi )\] equals [IIT 1997 Re-Exam; DCE 2001; UPSEAT 2001; Pb. CET 2002]
  A)
                  \[g(x)+g(\pi )\]  done clear
  B)
                  \[g(x)-g(\pi )\] done clear
  C)
                  \[g(x)g(\pi )\]    done clear
  D)
                  \[g(x)/g(\pi )\] done clear
  View Solution play_arrow

• question_answer131)
  The value of \[\int_{0}^{1}{(1+{{e}^{-{{x}^{2}}}})}\,dx=\]                              [IIT 1981]
  A)
                  \[-1\]     done clear
  B)
                  2 done clear
  C)
                  \[1+{{e}^{-1}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer132)
  \[\int_{0}^{2a}{\frac{f(x)}{f(x)+f(2a-x)}\,dx=}\]    [IIT 1988; Karnataka CET 2000]
  A)
                  \[a\]      done clear
  B)
                  \[\frac{a}{2}\] done clear
  C)
                  \[2a\]    done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer133)
  The value of \[\int_{0}^{n\pi +v}{|\sin x|\,dx}\] is                                           [IIT 1994]
  A)
                  x\[2n+1+\cos v\]              done clear
  B)
                  \[2n+1-\cos v\] done clear
  C)
                  \[2n+1\]               done clear
  D)
                  \[2n+\cos v\] done clear
  View Solution play_arrow

• question_answer134)
  If \[{{u}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}x\,dx,}\] then \[{{u}_{n}}+{{u}_{n-2}}=\]       [UPSEAT 2002]
  A)
                  \[\frac{1}{n-1}\]               done clear
  B)
                  \[\frac{1}{n+1}\] done clear
  C)
                  \[\frac{1}{2n-1}\] done clear
  D)
                  \[\frac{1}{2n+1}\] done clear
  View Solution play_arrow

• question_answer135)
  \[\int_{0}^{1}{\log \sin \left( \frac{\pi }{2}x \right)}\,dx=\]           [RPET 1997]
  A)
                  \[-\log 2\]            done clear
  B)
                  \[\log 2\] done clear
  C)
                  \[\frac{\pi }{2}\log 2\]    done clear
  D)
                  \[-\frac{\pi }{2}\log 2\] done clear
  View Solution play_arrow

• question_answer136)
  \[\int_{0}^{\pi /2}{{{\left( \frac{\theta }{\sin \theta } \right)}^{2}}d\theta =}\]
  A)
                  \[\pi \log 2\]       done clear
  B)
                  \[\frac{\pi }{\log 2}\] done clear
  C)
                  \[\pi \]  done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer137)
  \[\int_{0}^{1}{\frac{\log x}{\sqrt{1-{{x}^{2}}}}\,dx=}\] [BIT Ranchi 1984]
  A)
                  \[\frac{\pi }{2}\log 2\]    done clear
  B)
                  \[\pi \log 2\] done clear
  C)
                  \[-\frac{\pi }{2}\log 2\]  done clear
  D)
                  \[-\pi \log 2\] done clear
  View Solution play_arrow

• question_answer138)
  \[\int_{0}^{\pi /2}{x\cot x\,dx}\] equals                                                [RPET 1997]
  A)
                  \[-\frac{\pi }{2}\log 2\]  done clear
  B)
                  \[\frac{\pi }{2}\log 2\] done clear
  C)
                  \[\pi \log 2\]       done clear
  D)
                  \[-\pi \log 2\] done clear
  View Solution play_arrow

• question_answer139)
  . The integral value \[\int_{-2}^{0}{\left[ {{x}^{3}}+3{{x}^{2}}+3x+3+(x+1)\cos (x+1) \right]\ dx}\] is           [IIT Screening 2005]
  A)
                  2              done clear
  B)
                  4 done clear
  C)
                  0              done clear
  D)
                  8              done clear
  View Solution play_arrow

• question_answer140)
  If \[\int_{\sin x}^{1}{{{t}^{2}}f(t)\ dt=1-\sin x}\],\[x\in \left( 0,\frac{\pi }{2} \right)\] then \[f\ \left( \frac{1}{\sqrt{3}} \right)\]  equal to                                            [IIT Screening 2005]
  A)
                  3              done clear
  B)
                  \[\frac{1}{3}\] done clear
  C)
                  \[\frac{1}{\sqrt{3}}\]      done clear
  D)
                  \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer141)
  \[\int_{0}^{2n\pi }{\left( |\sin x|-\left. \left| \frac{1}{2}\sin x \right. \right| \right)}\ dx\] equals  [Orissa JEE 2005]
  A)
                  n             done clear
  B)
                  2n done clear
  C)
                  ?2n        done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer142)
  The value of  \[\int_{-a}^{a}{\frac{1}{x+{{x}^{3}}}dx}\] is                                               [AMU 2005]
  A)
                  0              done clear
  B)
                  \[\int_{0}^{a}{\frac{1}{1+{{x}^{6}}}\ }dx\] done clear
  C)
                  \[2\int_{0}^{a}{\frac{1}{1+{{x}^{3}}}\ }dx\]          done clear
  D)
                  \[\int_{0}^{a}{\frac{1}{1+{{(a-x)}^{3}}}\ }dx\] done clear
  View Solution play_arrow

• question_answer143)
  \[\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\tan x}}=}\]                                      [Kerala (Engg.) 2005]
  A)
                  \[\pi /12\]           done clear
  B)
                  \[\pi /2\] done clear
  C)
                  \[\pi /6\]              done clear
  D)
                  \[\pi /4\] done clear
  E)
                  \[2\pi /3\] done clear
  View Solution play_arrow

• question_answer144)
  \[\int_{\ -\pi }^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}\ dx}=\]                                   [Kerala (Engg.) 2005]
  A)
                  \[\pi /4\]              done clear
  B)
                  \[\pi /2\] done clear
  C)
                  \[3\pi /2\]           done clear
  D)
                  \[2\pi \] done clear
  E)
                  \[\pi \] done clear
  View Solution play_arrow

• question_answer145)
  If f is continuous function, then                 [Kerala (Engg.) 2005]
  A)
                  \[\int_{-2}^{2}{f(x)dx=\int_{0}^{2}{[f(x)-f(-x)]dx}}\]       done clear
  B)
                  \[\int_{-3}^{5}{2f(x)dx=\int_{-6}^{10}{f(x-1)dx}}\]           done clear
  C)
                  \[\int_{-3}^{5}{f(x)dx=\int_{-4}^{4}{f(x-1)dx}}\]               done clear
  D)
                  \[\int_{-3}^{5}{f(x)dx=\int_{-2}^{6}{f(x-1)dx}}\] done clear
  E)
                  \[\int_{-3}^{5}{f(x)dx=\int_{-6}^{10}{f(x/2)]dx}}\] done clear
  View Solution play_arrow