-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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