JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

done Integration of Rational Function by Using Partial Fractions Total Questions - 40

• question_answer1)
  \[\int_{{}}^{{}}{\frac{dx}{(x-{{x}^{2}})}=}\]           [Roorkee 1982]
  A)
                  \[\log x-\log (1-x)+c\] done clear
  B)
                  \[\log (1-{{x}^{2}})+c\] done clear
  C)
                  \[-\log x+\log (1-x)+c\] done clear
  D)
                  \[\log (x-{{x}^{2}})+c\] done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{{}}^{{}}{\frac{dx}{1+x+{{x}^{2}}+{{x}^{3}}}=}\]    [MP PET 1991]
  A)
                  \[\log \sqrt{1+x}-\frac{1}{2}\log \sqrt{1+{{x}^{2}}}+\frac{1}{2}{{\tan }^{-1}}x+c\] done clear
  B)
                  \[\log \sqrt{1+x}-\log \sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\] done clear
  C)
                  \[\log \sqrt{1+{{x}^{2}}}-\log \sqrt{1+x}+\frac{1}{2}{{\tan }^{-1}}x+c\] done clear
  D)
                  \[\log \sqrt{1+x}+{{\tan }^{-1}}x+\log \sqrt{1+{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer3)
  \[\int_{{}}^{{}}{\frac{x-1}{(x-3)(x-2)}dx=}\]          [Roorkee 1978]
  A)
                  \[\log (x-3)-\log (x-2)+c\]             done clear
  B)
                  \[\log {{(x-3)}^{2}}-\log (x-2)+c\] done clear
  C)
                  \[\log (x-3)+\log (x-2)+c\]            done clear
  D)
                  \[\log {{(x-3)}^{2}}+\log (x-2)+c\] done clear
  View Solution play_arrow

• question_answer4)
  \[\int_{{}}^{{}}{\frac{1}{\cos x(1+\cos x)}}\ dx=\]
  A)
                  \[\log (\sec x+\tan x)+2\tan \frac{x}{2}+c\] done clear
  B)
                  \[\log (\sec x+\tan x)-2\tan \frac{x}{2}+c\] done clear
  C)
                  \[\log (\sec x+\tan x)+\tan \frac{x}{2}+c\]            done clear
  D)
                  \[\log (\sec x+\tan x)-\tan \frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer5)
  \[\int_{{}}^{{}}{\frac{dx}{(x+1)(x+2)}=}\]                 [MP PET 1987]
  A)
                  \[\log \frac{x+2}{x+1}+c\] done clear
  B)
                  \[\log (x+1)+\log (x+2)+c\] done clear
  C)
                  \[\log \frac{x+1}{x+2}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer6)
  Correct evaluation of \[\int_{{}}^{{}}{\frac{x}{(x-2)(x-1)}\ dx}\] is              [MP PET 1993]
  A)
                  \[{{\log }_{e}}\frac{{{(x-2)}^{2}}}{(x-1)}+p\] done clear
  B)
                  \[{{\log }_{e}}\frac{(x-1)}{(x-2)}+p\] done clear
  C)
                  \[\frac{x-1}{x-2}+p\] done clear
  D)
                  \[2{{\log }_{e}}\left( \frac{x-2}{x-1} \right)+p\] (where p is an arbitrary constant) done clear
  View Solution play_arrow

• question_answer7)
  \[\int_{{}}^{{}}{\frac{1}{(x-1)({{x}^{2}}+1)}dx}=\]    [Roorkee 1984]
  A)
                  \[\frac{1}{2}\log (x-1)-\frac{1}{4}\log ({{x}^{2}}+1)-\frac{1}{2}{{\tan }^{-1}}x+c\]                done clear
  B)
                  \[\frac{1}{2}\log (x-1)+\frac{1}{4}\log ({{x}^{2}}+1)-\frac{1}{2}{{\tan }^{-1}}x+c\] done clear
  C)
                  \[\frac{1}{2}\log (x-1)-\frac{1}{2}\log ({{x}^{2}}+1)-\frac{1}{2}{{\tan }^{-1}}x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}+x-1}{{{x}^{2}}+x-6}\ dx=}\]     [AISSE 1988]
  A)
                  \[x+\log (x+3)+\log (x-2)+c\]      done clear
  B)
                  \[x-\log (x+3)+\log (x-2)+c\] done clear
  C)
                  \[x-\log (x+3)-\log (x-2)+c\]        done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}}{({{x}^{2}}+2)({{x}^{2}}+3)}\ }dx=\]        [AISSE 1990]
  A)
                  \[-\sqrt{2}{{\tan }^{-1}}x+\sqrt{3}{{\tan }^{-1}}x+c\] done clear
  B)
                  \[-\sqrt{2}{{\tan }^{-1}}\frac{x}{\sqrt{2}}+\sqrt{3}{{\tan }^{-1}}\frac{x}{\sqrt{3}}+c\] done clear
  C)
                  \[\sqrt{2}{{\tan }^{-1}}\frac{x}{\sqrt{2}}+\sqrt{3}{{\tan }^{-1}}\frac{x}{\sqrt{3}}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer10)
  \[\int_{{}}^{{}}{\frac{dx}{({{x}^{2}}+1)({{x}^{2}}+4)}=}\]                [MP PET 1995]
  A)
                  \[\frac{1}{3}{{\tan }^{-1}}x-\frac{1}{3}{{\tan }^{-1}}\frac{x}{2}+c\]            done clear
  B)
                  \[\frac{1}{3}{{\tan }^{-1}}x+\frac{1}{3}{{\tan }^{-1}}\frac{x}{2}+c\] done clear
  C)
                  \[\frac{1}{3}{{\tan }^{-1}}x-\frac{1}{6}{{\tan }^{-1}}\frac{x}{2}+c\]            done clear
  D)
                  \[{{\tan }^{-1}}x-2{{\tan }^{-1}}\frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{{}}^{{}}{\frac{1}{x-{{x}^{3}}}\ dx=}\]         [MP PET 1996]
  A)
                  \[\frac{1}{2}\log \frac{(1-{{x}^{2}})}{{{x}^{2}}}+c\] done clear
  B)
                  \[\log \frac{(1-x)}{x(1+x)}+c\] done clear
  C)
                  \[\log x(1-{{x}^{2}})+c\] done clear
  D)
                  \[\frac{1}{2}\log \frac{{{x}^{2}}}{(1-{{x}^{2}})}+c\] done clear
  View Solution play_arrow

• question_answer12)
  If \[\int_{{}}^{{}}{\sin 5x\cos 3x\ dx=-\frac{\cos 8x}{16}}+A\], then \[A=\]            [MP PET 1992]
  A)
                  \[\frac{\sin 2x}{16}+\]constant done clear
  B)
                  \[-\frac{\cos 2x}{4}+\]constant done clear
  C)
                  Constant done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{{}}^{{}}{{{\sin }^{3}}x{{\cos }^{2}}x\ dx=}\]
  A)
                  \[\frac{{{\cos }^{5}}x}{5}-\frac{{{\cos }^{3}}x}{3}+c\] done clear
  B)
                  \[\frac{{{\cos }^{5}}x}{5}+\frac{{{\cos }^{3}}x}{3}+c\] done clear
  C)
                  \[\frac{{{\sin }^{5}}x}{5}-\frac{{{\sin }^{3}}x}{3}+c\] done clear
  D)
                  \[\frac{{{\sin }^{5}}x}{5}+\frac{{{\sin }^{3}}x}{3}+c\] done clear
  View Solution play_arrow

• question_answer14)
  \[\int_{{}}^{{}}{\sin 2x\cos 3x\ dx=}\]    [Roorkee 1976]
  A)
                  \[\frac{1}{2}\left( \cos x+\frac{1}{5}\cos 5x \right)+c\] done clear
  B)
                  \[\frac{1}{2}\left( \cos x-\frac{1}{5}\cos 5x \right)+c\] done clear
  C)
                  \[\cos x+\frac{1}{5}\cos 5x+c\] done clear
  D)
                  \[\cos x-\frac{1}{5}\cos 5x+c\] done clear
  View Solution play_arrow

• question_answer15)
  \[\int_{{}}^{{}}{\frac{\cos x}{(1+\sin x)(2+\sin x)}\ dx=}\]       [Roorkee 1979]
  A)
                  \[\log [(1+\sin x)(2+\sin x)]+c\] done clear
  B)
                  \[\log \frac{2+\sin x}{1+\sin x}+c\] done clear
  C)
                  \[\log \frac{1+\sin x}{2+\sin x}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer16)
  \[\int_{{}}^{{}}{\frac{{{e}^{x}}}{(1+{{e}^{x}})(2+{{e}^{x}})}dx=}\]
  A)
                  \[\log [(1+{{e}^{x}})(2+{{e}^{x}})]+c\] done clear
  B)
                  \[\log \left[ \frac{1+{{e}^{x}}}{2+{{e}^{x}}} \right]+c\] done clear
  C)
                  \[\log [(1+{{e}^{x}})\sqrt{2+{{e}^{x}}}]+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}+1-2{{e}^{-x}}}=}\]
  A)
                  \[\log ({{e}^{x}}-1)-\log ({{e}^{x}}+2)+c\]             done clear
  B)
                  \[\frac{1}{2}\log ({{e}^{x}}-1)-\frac{1}{3}\log ({{e}^{x}}+2)+c\] done clear
  C)
                  \[\frac{1}{3}\log ({{e}^{x}}-1)-\frac{1}{3}\log ({{e}^{x}}+2)+c\] done clear
  D)
                  \[\frac{1}{3}\log ({{e}^{x}}-1)+\frac{1}{3}\log ({{e}^{x}}+2)+c\] done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{{}}^{{}}{\frac{x}{{{x}^{4}}-1}dx=}\]
  A)
                  \[\frac{1}{4}\log \left[ \frac{{{x}^{2}}-1}{{{x}^{2}}+1} \right]+c\] done clear
  B)
                  \[\frac{1}{4}\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right]+c\] done clear
  C)
                  \[\frac{1}{2}\log \left[ \frac{{{x}^{2}}-1}{{{x}^{2}}+1} \right]+c\] done clear
  D)
                  \[\frac{1}{2}\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right]+c\] done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{{}}^{{}}{{{\sin }^{5}}x{{\cos }^{4}}x\ dx=}\]
  A)
                  \[-\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\] done clear
  B)
                  \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\] done clear
  C)
                  \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x+\frac{1}{9}{{\cos }^{9}}x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer20)
  \[\int_{{}}^{{}}{\sqrt{{{x}^{2}}-8x+7}}\ dx=\]
  A)
                  \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}+9\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\]       done clear
  B)
                  \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}-3\sqrt{2}\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\] done clear
  C)
                  \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}-\frac{9}{2}\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\]     done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer21)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{2x-{{x}^{2}}}}=}\]              [MP PET 1991; Karnataka CET 2002]
  A)
                  \[{{\cos }^{-1}}(x-1)+c\] done clear
  B)
                  \[{{\sin }^{-1}}(x-1)+c\] done clear
  C)
                  \[{{\cos }^{-1}}(1+x)+c\] done clear
  D)
                  \[{{\sin }^{-1}}(1-x)+c\] done clear
  View Solution play_arrow

• question_answer22)
  \[\int_{{}}^{{}}{\frac{x\ dx}{({{x}^{2}}-{{a}^{2}})({{x}^{2}}-{{b}^{2}})}=}\]              [Roorkee 1976]
  A)
                  \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\] done clear
  B)
                  \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\] done clear
  C)
                  \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\]           done clear
  D)
                  \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\] done clear
  View Solution play_arrow

• question_answer23)
  \[\int_{{}}^{{}}{\frac{dx}{5+4\cos x}=}\]                 [Roorkee 1983; RPET 1997]
  A)
                  \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan x \right)+c\] done clear
  B)
                  \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan x \right)+c\] done clear
  C)
                  \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan \frac{x}{2} \right)+c\] done clear
  D)
                  \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan \frac{x}{2} \right)+c\] done clear
  View Solution play_arrow

• question_answer24)
  \[\int_{{}}^{{}}{\frac{1}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}dx=}\]
  A)
                  \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right) \right]+c\] done clear
  B)
                  \[\frac{1}{({{b}^{2}}-{{a}^{2}})}\left[ \frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right) \right]+c\] done clear
  C)
                  \[\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)+c\] done clear
  D)
                  \[\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)-\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)+c\] done clear
  View Solution play_arrow

• question_answer25)
  \[\int_{{}}^{{}}{\frac{1}{1+{{\cos }^{2}}x}dx}=\]
  A)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\tan x)+c\] done clear
  B)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{1}{2}\tan x \right)+c\] done clear
  C)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{2}}\tan x \right)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer26)
  \[\int_{{}}^{{}}{\frac{dx}{1+3{{\sin }^{2}}x}=}\] [Roorkee 1989; DCE 2001]
  A)
                  \[\frac{1}{3}{{\tan }^{-1}}(3{{\tan }^{2}}x)+c\] done clear
  B)
                  \[\frac{1}{2}{{\tan }^{-1}}(2\tan x)+c\] done clear
  C)
                  \[{{\tan }^{-1}}(\tan x)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer27)
  \[\int_{{}}^{{}}{\frac{dx}{2{{x}^{2}}+x+1}}\ \]equals        [RPET 1997]
  A)
                  \[\frac{1}{\sqrt{7}}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] done clear
  B)
                  \[\frac{1}{2\sqrt{7}}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] done clear
  C)
                  \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer28)
  \[\int{\frac{dx}{7+5\cos x}=}\]               [EAMCET 2002]
  A)
                  \[\frac{1}{\sqrt{6}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{6}}\tan \frac{x}{2} \right)+c\] done clear
  B)
                  \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] done clear
  C)
                  \[\frac{1}{4}{{\tan }^{-1}}\left( \tan \frac{x}{2} \right)+c\] done clear
  D)
                  \[\frac{1}{7}{{\tan }^{-1}}\left( \tan \frac{x}{2} \right)+c\] done clear
  View Solution play_arrow

• question_answer29)
  \[\int{\frac{dx}{{{x}^{2}}+4x+13}}\] is equal to                                     [Kerala (Engg.) 2002]
  A)
                  \[\log ({{x}^{2}}+4x+13)+c\] done clear
  B)
                  \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{x+2}{3} \right)+c\] done clear
  C)
                  \[\log (2x+4)+c\] done clear
  D)
                  \[\frac{2x+4}{{{({{x}^{2}}+4x+13)}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{{}}^{{}}{\frac{dx}{\cos x-\sin x}}\] is equal to                                           [AIEEE 2004]
  A)
                  \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right)\, \right|+c\] done clear
  B)
                   \[\frac{1}{\sqrt{2}}\log \left| \cot \left( \frac{x}{2} \right)\, \right|+c\] done clear
  C)
                  \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{3\pi }{8} \right)\, \right|+c\] done clear
  D)
                  \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{\pi }{8} \right)\, \right|+c\] done clear
  View Solution play_arrow

• question_answer31)
  \[\int_{{}}^{{}}{\frac{dx}{{{x}^{2}}+2x+2}=}\]                                        [Karnataka CET 2004]
  A)
                  \[{{\sin }^{-1}}(x+1)+c\] done clear
  B)
                  \[{{\sinh }^{-1}}(x+1)+c\] done clear
  C)
                  \[{{\tanh }^{-1}}(x+1)+c\] done clear
  D)
                  \[{{\tan }^{-1}}(x+1)+c\] done clear
  View Solution play_arrow

• question_answer32)
  \[\int_{{}}^{{}}{\frac{3\sin x+2\cos x}{3\cos x+2\sin x}\ dx=}\]
  A)
                  \[\frac{12}{13}x-\frac{5}{13}\log (3\cos x+2\sin x)\] done clear
  B)
                  \[\frac{12}{13}x+\frac{5}{13}\log (3\cos x+2\sin x)\] done clear
  C)
                  \[\frac{13}{12}x+\frac{5}{13}\log (3\cos x+2\sin x)\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer33)
  \[\int_{{}}^{{}}{\frac{dx}{x[{{(\log x)}^{2}}+4\log x-1]}}=\]
  A)
                  \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\] done clear
  B)
                  \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\] done clear
  C)
                  \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\] done clear
  D)
                  \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\] done clear
  View Solution play_arrow

• question_answer34)
  \[\int_{{}}^{{}}{\frac{dx}{x({{x}^{n}}+1)}=}\]          [Roorkee 1979]
  A)
                  \[n\log \frac{{{x}^{n}}}{{{x}^{n}}+1}+c\] done clear
  B)
                  \[n\log \frac{{{x}^{n}}+1}{{{x}^{n}}}+c\] done clear
  C)
                  \[\frac{1}{n}\log \frac{{{x}^{n}}}{{{x}^{n}}+1}+c\] done clear
  D)
                  \[\frac{1}{n}\log \frac{{{x}^{n}}+1}{{{x}^{n}}}+c\] done clear
  View Solution play_arrow

• question_answer35)
  \[\int_{{}}^{{}}{\frac{dx}{x({{x}^{7}}+1)}}=\]          [Karnataka CET 2004]
  A)
                  \[\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\] done clear
  B)
                  \[\frac{1}{7}\log \left( \frac{{{x}^{7}}}{{{x}^{7}}+1} \right)+c\] done clear
  C)
                  \[\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\] done clear
  D)
                  \[\frac{1}{7}\log \left( \frac{{{x}^{7}}+1}{{{x}^{7}}} \right)+c\] done clear
  View Solution play_arrow

• question_answer36)
  \[\int_{{}}^{{}}{\frac{dx}{x({{x}^{5}}+1)}}=\]          [UPSEAT 2004]
  A)
                  \[\frac{1}{5}\log {{x}^{5}}({{x}^{5}}+1)+c\] done clear
  B)
                  \[\frac{1}{5}\log {{x}^{5}}\left( \frac{1+{{x}^{5}}}{{{x}^{5}}} \right)+c\]              done clear
  C)
                  \[\frac{1}{5}\log {{x}^{5}}\left( \frac{{{x}^{5}}}{{{x}^{5}}+1} \right)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{{{x}^{4}}+1}dx=}\]           [AISSE 1990]
  A)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{2x} \right)+c\] done clear
  B)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{\sqrt{2x}} \right)+c\] done clear
  C)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{2\sqrt{x}} \right)+c\] done clear
  D)
                  \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{{{x}^{2}}-1}{\sqrt{2}x} \right)+c\] done clear
  View Solution play_arrow

• question_answer38)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}-1}{{{x}^{4}}+{{x}^{2}}+1}\ dx=}\]         [AISSE 1990]
  A)
                  \[\frac{1}{2}\log \left( \frac{{{x}^{2}}+x+1}{{{x}^{2}}-x+1} \right)+c\] done clear
  B)
                  \[\frac{1}{2}\log \left( \frac{{{x}^{2}}-x-1}{{{x}^{2}}+x+1} \right)+c\] done clear
  C)
                  \[\log \left( \frac{{{x}^{2}}-x+1}{{{x}^{2}}+x+1} \right)+c\] done clear
  D)
                  \[\frac{1}{2}\log \left( \frac{{{x}^{2}}-x+1}{{{x}^{2}}+x+1} \right)+c\] done clear
  View Solution play_arrow

• question_answer39)
  \[\int{\frac{x\,\,dx}{{{x}^{2}}+4x+5}=}\]                [RPET 2002]
  A)
                  \[\frac{1}{2}\log ({{x}^{2}}+4x+5)+2{{\tan }^{-1}}(x)+c\] done clear
  B)
                  \[\frac{1}{2}\log ({{x}^{2}}+4x+5)-{{\tan }^{-1}}(x+2)+c\] done clear
  C)
                  \[\frac{1}{2}\log ({{x}^{2}}+4x+5)+{{\tan }^{-1}}(x+2)+c\] done clear
  D)
                  \[\frac{1}{2}\log ({{x}^{2}}+4x+5)-2{{\tan }^{-1}}(x+2)+c\] done clear
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• question_answer40)
  If \[\int{\frac{2{{x}^{2}}+3.dx}{({{x}^{2}}-1)({{x}^{2}}-4)}}=\log {{\left( \frac{x-2}{x+2} \right)}^{a}}{{\left( \frac{x+1}{x-1} \right)}^{b}}+c\] then the values of a and b respectively are          [AMU 2005]
  A)
                  \[\frac{11}{12},\frac{5}{6}\] done clear
  B)
                  \[\frac{11}{12},\frac{-5}{6}\] done clear
  C)
                  \[-\frac{11}{12},\frac{5}{6}\] done clear
  D)
                  None of these done clear
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