JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

done Critical Thinking Total Questions - 40

• question_answer1)
  \[\int_{{}}^{{}}{\frac{dx}{\cos (x-a)\cos (x-b)}=}\]                
  A)
  \[\text{cosec}\,\,(a-b)\log \frac{\sin (x-a)}{\sin (x-b)}+c\] done clear
  B)
  \[\text{cosec}(a-b)\log \frac{\cos (x-a)}{\cos (x-b)}+c\] done clear
  C)
  \[\text{cosec}(a-b)\log \frac{\sin (x-b)}{\sin (x-a)}+c\]        done clear
  D)
  \[\text{cosec}(a-b)\log \frac{\cos (x-b)}{\cos (x-a)}+c\] done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x+a}+\sqrt{x+b}}}=\]            [AISSE 1989]
  A)
  \[\frac{2}{3(b-a)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}]+c\]         done clear
  B)
  \[\frac{2}{3(a-b)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}]+c\] done clear
  C)
  \[\frac{2}{3(a-b)}[{{(x+a)}^{3/2}}+{{(x+b)}^{3/2}}]+c\]        done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  \[\int_{{}}^{{}}{\frac{3\cos x+3\sin x}{4\sin x+5\cos x}\ dx=}\]            [EAMCET 1991]
  A)
  \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x+5\cos x)\]           done clear
  B)
  \[\frac{27}{41}x+\frac{3}{41}\log (4\sin x+5\cos x)\] done clear
  C)
  \[\frac{27}{41}x-\frac{3}{41}\log (4\sin x-5\cos x)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  If \[\int_{{}}^{{}}{(\sin 2x+\cos 2x)\ dx=\frac{1}{\sqrt{2}}\sin (2x-c)+a}\], then the value of a and c is             [Roorkee 1978]
  A)
  \[c=\pi /4\] and \[a=k\] (an arbitrary constant) done clear
  B)
  \[c=-\pi /4\] and \[a=\pi /2\] done clear
  C)
  \[c=\pi /2\] and a is an arbitrary constant            done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  \[\int_{{}}^{{}}{\frac{{{x}^{3}}-x-2}{(1-{{x}^{2}})}\ dx=}\]    [AI CBSE 1985]
  A)
  \[\log \left( \frac{x+1}{x-1} \right)-\frac{{{x}^{2}}}{2}+c\] done clear
  B)
  \[\log \left( \frac{x-1}{x+1} \right)+\frac{{{x}^{2}}}{2}+c\] done clear
  C)
  \[\log \left( \frac{x+1}{x-1} \right)+\frac{{{x}^{2}}}{2}+c\] done clear
  D)
  \[\log \left( \frac{x-1}{x+1} \right)-\frac{{{x}^{2}}}{2}+c\] done clear
  View Solution play_arrow

• question_answer6)
  \[\int_{{}}^{{}}{\frac{{{\sin }^{8}}x-{{\cos }^{8}}x}{1-2{{\sin }^{2}}x{{\cos }^{2}}x}\ dx=}\]   [IIT 1986]
  A)
  \[\sin 2x+c\] done clear
  B)
  \[-\frac{1}{2}\sin 2x+c\] done clear
  C)
  \[\frac{1}{2}\sin 2x+c\] done clear
  D)
  \[-\sin 2x+c\] done clear
  View Solution play_arrow

• question_answer7)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}dx}{{{(a+bx)}^{2}}}}=\]            [IIT 1979]
  A)
  \[\frac{1}{{{b}^{2}}}\left[ x+\frac{2a}{b}\log (a+bx)-\frac{{{a}^{2}}}{b}\frac{1}{a+bx} \right]\] done clear
  B)
  \[\frac{1}{{{b}^{2}}}\left[ x-\frac{2a}{b}\log (a+bx)+\frac{{{a}^{2}}}{b}\frac{1}{a+bx} \right]\] done clear
  C)
  \[\frac{1}{{{b}^{2}}}\left[ x+\frac{2a}{b}\log (a+bx)+\frac{{{a}^{2}}}{b}\frac{1}{a+bx} \right]\] done clear
  D)
  \[\frac{1}{{{b}^{2}}}\left[ x+\frac{a}{b}-\frac{2a}{b}\log (a+bx)-\frac{{{a}^{2}}}{b}\frac{1}{a+bx} \right]\] done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{{}}^{{}}{\frac{dx}{(1+{{x}^{2}})\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}}}=\]
  A)
  \[\frac{1}{q}\log [q{{\tan }^{-1}}x+\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}]+c\] done clear
  B)
  \[\log [q{{\tan }^{-1}}x+\sqrt{{{p}^{2}}+{{q}^{2}}{{({{\tan }^{-1}}x)}^{2}}}]+c\] done clear
  C)
  \[\frac{2}{3q}{{({{p}^{2}}+{{q}^{2}}{{\tan }^{-1}}x)}^{3/2}}+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{{}}^{{}}{\frac{{{x}^{5}}}{\sqrt{1+{{x}^{3}}}}dx=}\]      [IIT 1985]
  A)
  \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+c\] done clear
  B)
  \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\] done clear
  C)
  \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}-\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  \[\int{\frac{dx}{\sin x-\cos x+\sqrt{2}}}\] equals  [MP PET 2002]
  A)
  \[-\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] done clear
  B)
  \[\frac{1}{\sqrt{2}}\tan \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] done clear
  D)
  \[-\frac{1}{\sqrt{2}}\cot \left( \frac{x}{2}+\frac{\pi }{8} \right)+c\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{{}}^{{}}{\frac{a\ dx}{b+c{{e}^{x}}}}=\]        [MP PET 1988; BIT Ranchi 1979]
  A)
  \[\frac{a}{b}\log \left( \frac{{{e}^{x}}}{b+c{{e}^{x}}} \right)+c\] done clear
  B)
  \[\frac{a}{b}\log \left( \frac{b+c{{e}^{x}}}{{{e}^{x}}} \right)+c\] done clear
  C)
  \[\frac{b}{a}\log \left( \frac{{{e}^{x}}}{b+c{{e}^{x}}} \right)+c\] done clear
  D)
  \[\frac{b}{a}\log \left( \frac{b+c{{e}^{x}}}{{{e}^{x}}} \right)+c\] done clear
  View Solution play_arrow

• question_answer12)
  \[\int_{{}}^{{}}{\sin \sqrt{x}}\ dx=\]       [Roorkee 1977]
  A)
  \[2[\sin \sqrt{x}-\cos \sqrt{x}]+c\]        done clear
  B)
  \[2[\sin \sqrt{x}-\sqrt{x}\cos \sqrt{x}]+c\] done clear
  C)
  \[2[\sin \sqrt{x}+\cos \sqrt{x}]+c\]        done clear
  D)
  \[2[\sin \sqrt{x}+\sqrt{x}\cos \sqrt{x}]+c\] done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}}{{{(9-{{x}^{2}})}^{3/2}}}\ dx=}\]
  A)
  \[\frac{x}{\sqrt{9-{{x}^{2}}}}-{{\sin }^{-1}}\frac{x}{3}+c\] done clear
  B)
  \[\frac{x}{\sqrt{9-{{x}^{2}}}}+{{\sin }^{-1}}\frac{x}{3}+c\] done clear
  C)
  \[{{\sin }^{-1}}\frac{x}{3}-\frac{x}{\sqrt{9-{{x}^{2}}}}+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  \[\int_{{}}^{{}}{x\sqrt{\frac{1-{{x}^{2}}}{1+{{x}^{2}}}}}\ dx=\]
  A)
  \[\frac{1}{2}[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{4}}}]+c\]  done clear
  B)
  \[\frac{1}{2}[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{2}}}]+c\] done clear
  C)
  \[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{4}}}+c\]      done clear
  D)
  \[{{\sin }^{-1}}{{x}^{2}}+\sqrt{1-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer15)
  If \[\int_{{}}^{{}}{f(x)\sin x\cos x\ dx=\frac{1}{2({{b}^{2}}-{{a}^{2}})}\log (f(x))}+c\], then \[f(x)=\]
  A)
  \[\frac{1}{{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x}\] done clear
  B)
  \[\frac{1}{{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x}\] done clear
  C)
  \[\frac{1}{{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}{{\sin }^{2}}x}\] done clear
  D)
  \[\frac{1}{{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}{{\sin }^{2}}x}\] done clear
  View Solution play_arrow

• question_answer16)
  \[\int_{{}}^{{}}{\frac{dx}{4{{\sin }^{2}}x+5{{\cos }^{2}}x}=}\]              [AISSE 1986]
  A)
  \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] done clear
  B)
  \[\frac{1}{\sqrt{5}}{{\tan }^{-1}}\left( \frac{\tan x}{\sqrt{5}} \right)+c\] done clear
  C)
  \[\frac{1}{2\sqrt{5}}{{\tan }^{-1}}\left( \frac{2\tan x}{\sqrt{5}} \right)+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}\ dx=}\]         [MP PET 1991]
  A)
  \[{{\tan }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] done clear
  B)
  \[{{\cot }^{-1}}\left( \frac{1+{{x}^{2}}}{x} \right)+c\] done clear
  C)
  \[{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] done clear
  D)
  \[{{\cot }^{-1}}\left( \frac{{{x}^{2}}-1}{x} \right)+c\] done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{{}}^{{}}{{{(\log x)}^{2}}\ dx=}\]  [IIT 1971, 77]
  A)
  \[x{{(\log x)}^{2}}-2x\log x-2x+c\] done clear
  B)
  \[x{{(\log x)}^{2}}-2x\log x-x+c\] done clear
  C)
  \[x{{(\log x)}^{2}}-2x\log x+2x+c\]         done clear
  D)
  \[x{{(\log x)}^{2}}-2x\log x+x+c\] done clear
  View Solution play_arrow

• question_answer19)
  The value of \[\int{\frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{x}dx}\] will be   [UPSEAT 1999]
  A)
  \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,-a{{\tan }^{-1}}\left[ \frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{a} \right]\] done clear
  B)
  \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,+a{{\tan }^{-1}}\left[ \frac{\sqrt{({{x}^{2}}-{{a}^{2}})}}{a} \right]\] done clear
  C)
  \[\sqrt{({{x}^{2}}-{{a}^{2}})}\,+{{a}^{2}}{{\tan }^{-1}}[\sqrt{{{x}^{2}}-{{a}^{2}}}]\] done clear
  D)
  \[{{\tan }^{-1}}x/a+c\] done clear
  View Solution play_arrow

• question_answer20)
  \[\int_{{}}^{{}}{{{\tan }^{3}}}2x\sec 2x\ dx=\]      [IIT 1977]
  A)
  \[\frac{1}{6}{{\sec }^{3}}2x-\frac{1}{2}\sec 2x+c\] done clear
  B)
  \[\frac{1}{6}{{\sec }^{3}}2x+\frac{1}{2}\sec 2x+c\] done clear
  C)
  \[\frac{1}{9}{{\sec }^{2}}2x-\frac{1}{3}\sec 2x+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  \[\int_{{}}^{{}}{x{{\sin }^{-1}}x\ dx}=\]  [MP PET 1991]
  A)
  \[\left( \frac{{{x}^{2}}}{2}-\frac{1}{4} \right){{\sin }^{-1}}x+\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] done clear
  B)
  \[\left( \frac{{{x}^{2}}}{2}+\frac{1}{4} \right){{\sin }^{-1}}x+\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] done clear
  C)
  \[\left( \frac{{{x}^{2}}}{2}-\frac{1}{4} \right){{\sin }^{-1}}x-\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] done clear
  D)
  \[\left( \frac{{{x}^{2}}}{2}+\frac{1}{4} \right){{\sin }^{-1}}x-\frac{x}{4}\sqrt{1-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer22)
  \[\int_{{}}^{{}}{\sqrt{\frac{a-x}{x}}\ dx=}\]
  A)
  \[a\left[ {{\sin }^{-1}}\sqrt{\frac{x}{a}}+\sqrt{\frac{x}{a}}\sqrt{\frac{a-x}{a}} \right]+c\] done clear
  B)
  \[{{\sin }^{-1}}\frac{x}{a}+\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}+c\] done clear
  C)
  \[a\left[ {{\sin }^{-1}}\frac{x}{a}-\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}} \right]+c\]               done clear
  D)
  \[{{\sin }^{-1}}\frac{x}{a}-\frac{x}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer23)
  If \[x\in \left( \frac{\pi }{4},\frac{3\pi }{4} \right)\], then \[\int_{{}}^{{}}{\frac{\sin x-\cos x}{\sqrt{1-\sin 2x}}{{e}^{\sin x}}\cos x\ dx=}\]
  A)
  \[{{e}^{\sin x}}+c\] done clear
  B)
  \[{{e}^{\sin x-\cos x}}+c\] done clear
  C)
  \[{{e}^{\sin x+\cos x}}+c\] done clear
  D)
  \[{{e}^{\cos x-\sin x}}+c\] done clear
  View Solution play_arrow

• question_answer24)
  If \[\int_{{}}^{{}}{\frac{4{{e}^{x}}+6{{e}^{-x}}}{9{{e}^{x}}-4{{e}^{-x}}}dx=Ax+B\log (9{{e}^{2x}}-4)}+C\], then A, B and C are           [IIT 1990]
  A)
  \[A=\frac{3}{2},\ B=\frac{36}{35},\ C=\frac{3}{2}\log 3+\]constant    done clear
  B)
  \[A=\frac{3}{2},\ B=\frac{35}{36},\ C=\frac{3}{2}\log 3+\]constant done clear
  C)
  \[A=-\frac{3}{2},\ B=-\frac{35}{36},\ C=-\frac{3}{2}\log 3+\]constant done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  The value of \[\int{{{\sec }^{3}}x\,\,dx}\] will be  [UPSEAT 1999]
  A)
  \[\frac{1}{2}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] done clear
  B)
  \[\frac{1}{3}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] done clear
  C)
  \[\frac{1}{4}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] done clear
  D)
  \[\frac{1}{8}\left[ \,\sec x\tan x+\log (\sec x+\tan x) \right]\] done clear
  View Solution play_arrow

• question_answer26)
  \[\int_{{}}^{{}}{\frac{x-1}{{{(x+1)}^{3}}}{{e}^{x}}\ dx=}\]       [IIT 1983; MP PET 1990]
  A)
  \[\frac{-{{e}^{x}}}{{{(x+1)}^{2}}}+c\] done clear
  B)
  \[\frac{{{e}^{x}}}{{{(x+1)}^{2}}}+c\] done clear
  C)
  \[\frac{{{e}^{x}}}{{{(x+1)}^{3}}}+c\] done clear
  D)
  \[\frac{-{{e}^{x}}}{{{(x+1)}^{3}}}+c\] done clear
  View Solution play_arrow

• question_answer27)
  If \[I=\int_{{}}^{{}}{{{e}^{x}}\sin 2x\ dx}\], then for what value of K, \[KI={{e}^{x}}(\sin 2x-2\cos 2x)+\]constant             [MP PET 1992]
  A)
  1 done clear
  B)
  3 done clear
  C)
  5 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer28)
  The value of \[\int{\frac{dx}{3-2x-{{x}^{2}}}}\] will be         [UPSEAT 1999]
  A)
  \[\frac{1}{4}\log \,\left( \frac{3+x}{1-x} \right)\] done clear
  B)
  \[\frac{1}{3}\log \,\left( \frac{3+x}{1-x} \right)\] done clear
  C)
  \[\frac{1}{2}\log \,\left( \frac{3+x}{1-x} \right)\] done clear
  D)
  \[\log \,\left( \frac{1-x}{3+x} \right)\] done clear
  View Solution play_arrow

• question_answer29)
  \[\int_{{}}^{{}}{x\sqrt{2x+3}}\ dx=\]     [AISSE 1985]
  A)
  \[\frac{x}{3}{{(2x+3)}^{3/2}}-\frac{1}{15}{{(2x+3)}^{5/2}}+c\] done clear
  B)
  \[\frac{x}{3}{{(2x+3)}^{3/2}}+\frac{1}{15}{{(2x+3)}^{5/2}}+c\] done clear
  C)
  \[\frac{x}{2}{{(2x+3)}^{3/2}}+\frac{1}{6}{{(2x+3)}^{5/2}}+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{{}}^{{}}{\cos 2\theta \log \left( \frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta } \right)\ d\theta =}\]       [IIT 1994]
  A)
  \[{{(\cos \theta -\sin \theta )}^{2}}\log \left( \frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta } \right)\] done clear
  B)
  \[{{(\cos \theta +\sin \theta )}^{2}}\log \left( \frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta } \right)\] done clear
  C)
  \[\frac{{{(\cos \theta -\sin \theta )}^{2}}}{2}\log \left( \frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta } \right)\] done clear
  D)
  \[\frac{1}{2}\sin 2\theta \log \tan \left( \frac{\pi }{4}+\theta  \right)-\frac{1}{2}\log \sec 2\theta \] done clear
  View Solution play_arrow

• question_answer31)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}}{{{(x\sin x+\cos x)}^{2}}}\ dx=}\]          [MNR 1989; RPET 2000]
  A)
  \[\frac{\sin x+\cos x}{x\sin x+\cos x}\] done clear
  B)
  \[\frac{x\sin x-\cos x}{x\sin x+\cos x}\] done clear
  C)
  \[\frac{\sin x-x\cos x}{x\sin x+\cos x}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  If \[u=\int_{{}}^{{}}{{{e}^{ax}}\cos bx\ dx}\] and \[v=\int_{{}}^{{}}{{{e}^{ax}}\sin bx\ dx}\], then \[({{a}^{2}}+{{b}^{2}})({{u}^{2}}+{{v}^{2}})=\]
  A)
  \[2{{e}^{ax}}\] done clear
  B)
  \[({{a}^{2}}+{{b}^{2}}){{e}^{2ax}}\] done clear
  C)
  \[{{e}^{2ax}}\] done clear
  D)
  \[({{a}^{2}}-{{b}^{2}}){{e}^{2ax}}\] done clear
  View Solution play_arrow

• question_answer33)
  If \[{{I}_{n}}=\int{{{(\log x)}^{n}}\,\,dx},\] then \[{{I}_{n}}+n{{I}_{n-1}}=\] [Karnataka CET 2003]
  A)
  \[x{{(\log x)}^{n}}\] done clear
  B)
  \[{{(x\log x)}^{n}}\] done clear
  C)
  \[{{(\log x)}^{n-1}}\] done clear
  D)
  \[n{{(\log x)}^{n}}\] done clear
  View Solution play_arrow

• question_answer34)
  \[\int_{{}}^{{}}{{{e}^{x/2}}\sin \left( \frac{x}{2}+\frac{\pi }{4} \right)\ dx=}\]    [Roorkee 1982]
  A)
  \[{{e}^{x/2}}\cos \frac{x}{2}+c\] done clear
  B)
  \[\sqrt{2}{{e}^{x/2}}\cos \frac{x}{2}+c\] done clear
  C)
  \[{{e}^{x/2}}\sin \frac{x}{2}+c\] done clear
  D)
  \[\sqrt{2}{{e}^{x/2}}\sin \frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer35)
  If \[\int_{{}}^{{}}{\frac{2x+3}{{{x}^{2}}-5x+6}}\ dx=9\ \ln (x-3)-7\ln (x-2)+A\], then \[A=\]                [MP PET 1992]
  A)
  \[5\ln (x-2)+\]Constant done clear
  B)
  \[-4\ln (x-3)+\]constant done clear
  C)
  Constant done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer36)
  \[\int_{{}}^{{}}{\frac{dx}{2+\cos x}=}\]
  A)
  \[2{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] done clear
  B)
  \[\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] done clear
  C)
  \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{{}}^{{}}{\frac{x}{{{x}^{4}}+{{x}^{2}}+1}dx}\] equal to                [MP PET 2004]
  A)
  \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{3} \right)\] done clear
  B)
  \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2{{x}^{2}}+1}{\sqrt{3}} \right)\] done clear
  C)
  \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}(2{{x}^{2}}+1)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer38)
  \[\int_{{}}^{{}}{\frac{dx}{(\sin x+\sin 2x)}=}\]       [IIT 1984]
  A)
  \[\frac{1}{6}\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\]         done clear
  B)
  \[6\log (1-\cos x)+2\log (1+\cos x)-\frac{2}{3}\log (1+2\cos x)\] done clear
  C)
  \[6\log (1-\cos x)+\frac{1}{2}\log (1+\cos x)+\frac{2}{3}\log (1+2\cos x)\]        done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  If \[\int_{{}}^{{}}{\frac{2x+3}{(x-1)({{x}^{2}}+1)}\ dx={{\log }_{e}}\left\{ {{(x-1)}^{\frac{5}{2}}}{{({{x}^{2}}+1)}^{a}} \right\}}-\frac{1}{2}{{\tan }^{-1}}x+A\]  , where A is any arbitrary constant, then the value of ?a? is  [MP PET 1998]
  A)
  5/4 done clear
  B)
  - 5/3 done clear
  C)
  - 5/6 done clear
  D)
  - 5/4 done clear
  View Solution play_arrow

• question_answer40)
  If \[\int{\frac{(2{{x}^{2}}+1)\,\,dx}{({{x}^{2}}-4)\,\,({{x}^{2}}-1)}=\log \left[ {{\left( \frac{x+1}{x-1} \right)}^{a}}\,\,{{\left( \frac{x-2}{x+2} \right)}^{b}} \right]}+C,\] then the values of a and b are respectively [Roorkee 2000]
  A)
  1/2, ¾ done clear
  B)
  -1, 3/2 done clear
  C)
  1, 3/2 done clear
  D)
  -1/2, ¾ done clear
  View Solution play_arrow