JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

done Integration by Parts Total Questions - 73

• question_answer1)
  \[\int_{{}}^{{}}{x{{\sec }^{2}}x\ dx}=\] [RPET 1996, 2003; MP PET 1987, 97; Pb. CET 2002]
  A)
                  \[\tan x+\log \cos x+c\] done clear
  B)
                  \[\frac{{{x}^{2}}}{2}{{\sec }^{2}}x+\log \cos x+c\] done clear
  C)
                  \[x\tan x+\log \sec x+c\] done clear
  D)
                  \[x\tan x+\log \cos x+c\] done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{{}}^{{}}{\sin (\log x)dx=}\]
  A)
                  \[\frac{1}{2}x[\cos (\log x)-\sin (\log x)]\] done clear
  B)
                  \[\cos (\log x)-x\] done clear
  C)
                  \[\frac{1}{2}x[\sin (\log x)-\cos (\log x)]\] done clear
  D)
                  \[-\cos \log x\] done clear
  View Solution play_arrow

• question_answer3)
  If \[\int_{{}}^{{}}{x\sin xdx=-x\cos x+A}\], then \[A=\] [MP PET 1992, 2000; RPET 1997]
  A)
                  \[\sin x+\]Constant done clear
  B)
                  \[\cos x+\]Constant done clear
  C)
                  Constant done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer4)
  \[\int_{{}}^{{}}{x\log xdx=}\]      [MP PET 1987]
  A)
                  \[\frac{{{x}^{2}}}{2}\log x-\frac{{{x}^{2}}}{2}+c\] done clear
  B)
                  \[\frac{{{x}^{2}}}{2}\log x-\frac{{{x}^{2}}}{4}+c\] done clear
  C)
                  \[\frac{{{x}^{2}}}{2}\log x+\frac{{{x}^{2}}}{2}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer5)
  \[\int_{{}}^{{}}{x\cos x\ dx=}\]  [MP PET 1988]
  A)
                  \[x\sin x+\cos x\] done clear
  B)
                  \[x\sin x-\cos x\] done clear
  C)
                  \[x\cos x+\sin x\] done clear
  D)
                  \[x\cos x-\sin x\] done clear
  View Solution play_arrow

• question_answer6)
  \[\int_{{}}^{{}}{x{{\cos }^{2}}}xdx=\]      [IIT 1972]
  A)
                  \[\frac{{{x}^{4}}}{4}-\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+c\] done clear
  B)
                  \[\frac{{{x}^{2}}}{4}+\frac{1}{4}x\sin 2x+\frac{1}{8}\cos 2x+c\] done clear
  C)
                  \[\frac{{{x}^{4}}}{4}-\frac{1}{4}x\sin 2x+\frac{1}{8}\cos 2x+c\] done clear
  D)
                  \[\frac{{{x}^{4}}}{4}+\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+c\] done clear
  View Solution play_arrow

• question_answer7)
  \[\int_{{}}^{{}}{{{\tan }^{-1}}x\,dx=}\]    [Roorkee 1977]
  A)
                  \[x{{\tan }^{-1}}x+\frac{1}{2}\log (1+{{x}^{2}})\] done clear
  B)
                  \[x{{\tan }^{-1}}x-\frac{1}{2}\log (1+{{x}^{2}})\] done clear
  C)
                  \[(x-1){{\tan }^{-1}}x\] done clear
  D)
                  \[x{{\tan }^{-1}}x-\log (1+{{x}^{2}})\] done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{{}}^{{}}{x{{\tan }^{-1}}}xdx=\]     [Roorkee 1979]
  A)
                  \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x-\frac{1}{2}x+c\] done clear
  B)
                  \[\frac{1}{2}({{x}^{2}}-1){{\tan }^{-1}}x-\frac{1}{2}x+c\] done clear
  C)
                  \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x+\frac{1}{2}x+c\]               done clear
  D)
                  \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x-x+c\] done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{{}}^{{}}{\left[ \log (\log x)+\frac{1}{{{(\log x)}^{2}}} \right]}\ dx=\]
  A)
                  \[x\log (\log x)+\frac{x}{\log x}+c\] done clear
  B)
                  \[x\log (\log x)-\frac{x}{\log x}+c\] done clear
  C)
                  \[x\log (\log x)+\frac{\log x}{x}+c\] done clear
  D)
                  \[x\log (\log x)-\frac{\log x}{x}+c\] done clear
  View Solution play_arrow

• question_answer10)
  \[\int_{{}}^{{}}{[\sin (\log x)+\cos (\log x)]}\ dx=\]           [MP PET 1991]
  A)
                  \[x\cos (\log x)+c\] done clear
  B)
                  \[\sin (\log x)+c\] done clear
  C)
                  \[\cos (\log x)+c\] done clear
  D)
                  \[x\sin (\log x)+c\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{{}}^{{}}{\frac{\log x\ dx}{{{x}^{3}}}=}\]    [Roorkee 1986]
  A)
                  \[\frac{1}{4{{x}^{2}}}(2\log x-1)+c\] done clear
  B)
                  \[-\frac{1}{4{{x}^{2}}}(2\log x+1)+c\] done clear
  C)
                  \[\frac{1}{4{{x}^{2}}}(2\log x+1)+c\] done clear
  D)
                  \[\frac{1}{4{{x}^{2}}}(1-2\log x)+c\] done clear
  View Solution play_arrow

• question_answer12)
  \[\int_{{}}^{{}}{{{e}^{-2x}}\sin 3x\ dx=}\]
  A)
                  \[\frac{1}{13}{{e}^{-2x}}[\sin 3x+\cos 3x]+c\]     done clear
  B)
                  \[-\frac{1}{13}{{e}^{-2x}}[\sin 3x+\cos 3x]+c\] done clear
  C)
                  \[\frac{1}{13}{{e}^{-2x}}[2\sin 3x+3\cos 3x]+c\] done clear
  D)
                  \[-\frac{1}{13}{{e}^{-2x}}[2\sin 3x+3\cos 3x]+c\] done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{{}}^{{}}{x\sin x{{\sec }^{3}}x\,dx=}\]
  A)
                  \[\frac{1}{2}[{{\sec }^{2}}x-\tan x]+c\] done clear
  B)
                  \[\frac{1}{2}[x{{\sec }^{2}}x-\tan x]+c\] done clear
  C)
                  \[\frac{1}{2}[x{{\sec }^{2}}x+\tan x]+c\] done clear
  D)
                  \[\frac{1}{2}[{{\sec }^{2}}x+\tan x]+c\] done clear
  View Solution play_arrow

• question_answer14)
  \[\int_{{}}^{{}}{x{{\sin }^{2}}x\ dx=}\]    [BIT Ranchi 1977; IIT 1972]
  A)
                  \[\frac{{{x}^{2}}}{4}+\frac{x}{4}\sin 2x+\frac{1}{8}\cos 2x+c\] done clear
  B)
                  \[\frac{{{x}^{2}}}{4}-\frac{x}{4}\sin 2x+\frac{1}{8}\cos 2x+c\] done clear
  C)
                  \[\frac{{{x}^{2}}}{4}+\frac{x}{4}\sin 2x-\frac{1}{8}\cos 2x+c\]      done clear
  D)
                  \[\frac{{{x}^{2}}}{4}-\frac{x}{4}\sin 2x-\frac{1}{8}\cos 2x+c\] done clear
  View Solution play_arrow

• question_answer15)
  \[\int_{{}}^{{}}{{{e}^{2x+\log x}}}dx=\]
  A)
                  \[\frac{1}{4}(2x-1)+\frac{2}{x+1}+c\] done clear
  B)
                  \[\frac{1}{4}(2x+1)+\frac{2}{x+1}+c\] done clear
  C)
                  \[\frac{1}{2}(2x+1){{e}^{2x}}+c\] done clear
  D)
                  \[\frac{1}{2}(2x+1){{e}^{2x}}+c\] done clear
  View Solution play_arrow

• question_answer16)
  \[\int_{{}}^{{}}{\log (x+1)dx=}\]              [Roorkee 1974]
  A)
                  \[(x+1)\log (x+1)-x+c\] done clear
  B)
                  \[(x+1)\log (x+1)+x+c\] done clear
  C)
                  \[(x-1)\log (x+1)-x+c\] done clear
  D)
                  \[(x-1)\log (x+1)+x+c\] done clear
  View Solution play_arrow

• question_answer17)
  If \[\int_{{}}^{{}}{\ln ({{x}^{2}}+x)dx=x\ln ({{x}^{2}}+x)+A}\], then \[A=\] [MP PET 1992]
  A)
                  \[2x+\ln (x+1)+\]constant done clear
  B)
                  \[2x-\ln (x+1)+\]constant done clear
  C)
                  Constant done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{{}}^{{}}{{{x}^{2}}\sin 2x}\ dx=\] [IIT 1974]
  A)
                  \[\frac{1}{2}{{x}^{2}}\cos 2x+\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\]  done clear
  B)
                  \[-\frac{1}{2}{{x}^{2}}\cos 2x+\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] done clear
  C)
                  \[\frac{1}{2}{{x}^{2}}\cos 2x-\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{{}}^{{}}{\log xdx=}\]        [MNR 1979; BIT Ranchi 1992; SCRA 1996]
  A)
                  \[x+x\log x+c\] done clear
  B)
                  \[x\log x-x+c\] done clear
  C)
                  \[{{x}^{2}}\log x+c\] done clear
  D)
                  \[\frac{1}{x}\log x+x+c\] done clear
  View Solution play_arrow

• question_answer20)
  \[\int_{{}}^{{}}{{{\log }_{10}}x\ dx=}\]    [Roorkee 1973]
  A)
                  \[x{{\log }_{10}}x+c\] done clear
  B)
                  \[x({{\log }_{10}}x+{{\log }_{10}}e)+c\] done clear
  C)
                  \[{{\log }_{10}}x+c\] done clear
  D)
                  \[x({{\log }_{10}}x-{{\log }_{10}}e)+c\] done clear
  View Solution play_arrow

• question_answer21)
  \[\int_{{}}^{{}}{\frac{1}{{{\log }_{x}}e}dx=}\]      [MP PET 1994]
  A)
                  \[\log {{\log }_{x}}e+c\] done clear
  B)
                  \[\frac{1}{{{({{\log }_{x}}e)}^{2}}}+c\] done clear
  C)
                  \[x\log \left( \frac{x}{e} \right)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer22)
  If \[\int_{{}}^{{}}{{{e}^{x}}\sin x\ dx=\frac{1}{2}{{e}^{x}}\ .\ a+c}\], then \[a=\] [MP PET 1989]
  A)
                  \[\sin x-\cos x\] done clear
  B)
                  \[\cos x-\sin x\] done clear
  C)
                  \[-\cos x-\sin x\] done clear
  D)
                  \[\cos x+\sin x\] done clear
  View Solution play_arrow

• question_answer23)
  \[\int_{{}}^{{}}{{{x}^{n}}\log x\ dx=}\]
  A)
                  \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x+\frac{1}{n+1} \right\}+c\] done clear
  B)
                  \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x+\frac{2}{n+1} \right\}+c\] done clear
  C)
                  \[\frac{{{x}^{n+1}}}{n+1}\left\{ 2\log x-\frac{1}{n+1} \right\}+c\] done clear
  D)
                  \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x-\frac{1}{n+1} \right\}+c\] done clear
  View Solution play_arrow

• question_answer24)
  \[\int_{{}}^{{}}{\log x(\log x+2)\ dx=}\]
  A)
                  \[x{{(\log x)}^{2}}+c\] done clear
  B)
                  \[x{{(1+\log x)}^{2}}+c\] done clear
  C)
                  \[x[1+{{(\log x)}^{2}}]+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

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

• question_answer26)
  \[\int_{{}}^{{}}{\frac{\log x}{{{(1+\log x)}^{2}}}dx=}\]
  A)
                  \[\frac{1}{1+\log x}+c\] done clear
  B)
                  \[\frac{x}{{{(1+\log x)}^{2}}}+c\] done clear
  C)
                  \[\frac{x}{1+\log x}+c\] done clear
  D)
                  \[\frac{1}{{{(1+\log x)}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer27)
  \[\int_{{}}^{{}}{\left( \frac{2+\sin 2x}{1+\cos 2x} \right)\,\,{{e}^{x}}dx=}\]           [AISSE 1982]
  A)
                  \[{{e}^{x}}\cot x+c\] done clear
  B)
                  \[-{{e}^{x}}\cot x+c\] done clear
  C)
                  \[-{{e}^{x}}\tan x+c\] done clear
  D)
                  \[{{e}^{x}}\tan x+c\] done clear
  View Solution play_arrow

• question_answer28)
  \[\int_{{}}^{{}}{{{e}^{x}}\sin x\ dx=}\]    [IIT 1978; AI CBSE 1980; MP PET 1999]
  A)
                  \[\frac{1}{2}{{e}^{x}}(\sin x+\cos x)+c\] done clear
  B)
                  \[\frac{1}{2}{{e}^{x}}(\sin x-\cos x)+c\] done clear
  C)
                  \[{{e}^{x}}(\sin x+\cos x)+c\] done clear
  D)
                  \[{{e}^{x}}(\sin x-\cos x)+c\] done clear
  View Solution play_arrow

• question_answer29)
  \[\int_{{}}^{{}}{(1-{{x}^{2}})\log x\ dx=}\]               [DSSE 1982]
  A)
                  \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x-\left( x-\frac{{{x}^{3}}}{9} \right)+c\] done clear
  B)
                  \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x+\left( x-\frac{{{x}^{3}}}{9} \right)+c\] done clear
  C)
                  \[\left( x+\frac{{{x}^{3}}}{3} \right)\log x+\left( x+\frac{{{x}^{3}}}{9} \right)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{{}}^{{}}{{{e}^{2x}}(-\sin x+2\cos x)\ dx=}\] [DSSE 1987]
  A)
                  \[{{e}^{2x}}\sin x+c\] done clear
  B)
                  \[-{{e}^{2x}}\sin x+c\] done clear
  C)
                  \[-{{e}^{2x}}\cos x+c\] done clear
  D)
                  \[{{e}^{2x}}\cos x+c\] done clear
  View Solution play_arrow

• question_answer31)
  \[\int_{{}}^{{}}{[f(x)g''(x)-f''(x)g(x)]}\ dx\] is equal to      [MP PET 2001]
  A)
                  \[\frac{f(x)}{g'(x)}\] done clear
  B)
                  \[f'(x)g(x)-f(x)g'(x)\] done clear
  C)
                  \[f(x)g'(x)-f'(x)g(x)\] done clear
  D)
                  \[f(x)g'(x)+f'(x)g(x)\] done clear
  View Solution play_arrow

• question_answer32)
  The value of \[\int_{{}}^{{}}{x\sin kx\ dx}\]is
  A)
                  \[\frac{\sin kx}{k}+c\] done clear
  B)
                  \[\frac{\cos kx}{k}+c\] done clear
  C)
                  \[\frac{\sin x}{k}+c\] done clear
  D)
                  \[-\frac{x\,\cos kx}{k}+\frac{\sin kx}{{{k}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer33)
  \[\int_{{}}^{{}}{x{{e}^{x}}\ dx}\]is equal to           [SCRA 1996]
  A)
                  \[(x+1){{e}^{x}}+c\] done clear
  B)
                  \[(x-1){{e}^{x}}+c\] done clear
  C)
                  \[x{{e}^{x}}+1+c\] done clear
  D)
                  \[x{{e}^{x}}-1+c\] done clear
  View Solution play_arrow

• question_answer34)
  \[\int_{{}}^{{}}{{{x}^{3}}{{e}^{{{x}^{2}}}}dx=}\]      [MNR 1980]
  A)
                  \[\frac{1}{2}({{x}^{2}}+1){{e}^{{{x}^{2}}}}+c\] done clear
  B)
                  \[({{x}^{2}}+1){{e}^{{{x}^{2}}}}+c\] done clear
  C)
                  \[\frac{1}{2}({{x}^{2}}-1){{e}^{{{x}^{2}}}}+c\] done clear
  D)
                  \[({{x}^{2}}-1){{e}^{{{x}^{2}}}}+c\] done clear
  View Solution play_arrow

• question_answer35)
  The value of \[\int{\frac{\log x}{{{(x+1)}^{2}}}dx}\] is      [UPSEAT 1999]
  A)
                  \[\frac{-\log x}{x+1}+\log x-\log \,(x+1)\] done clear
  B)
                  \[\frac{\log x}{\left( x+1 \right)}+\log x-\log \,(x+1)\] done clear
  C)
                  \[\frac{\log x}{x+1}-\log x-\log \,(x+1)\]                done clear
  D)
                  \[\frac{-\log x}{x+1}-\log x-\log \,(x+1)\] done clear
  View Solution play_arrow

• question_answer36)
  If \[\int{x{{e}^{2x}}\,\,dx}\] is equal to \[{{e}^{2x}}f(x)+C\] where C is constant of integration, then f(x) is                 [UPSEAT 2001]
  A)
                  \[(3x-1)/4\] done clear
  B)
                  \[(2x+1)/2\] done clear
  C)
                  \[(2x-1)/4\] done clear
  D)
                  \[(x-4)/6\] done clear
  View Solution play_arrow

• question_answer37)
  If \[\frac{d}{dx}f(x)=x\cos x+\sin x\] and \[f(0)=2\], then \[f(x)=\] [MP PET 1989]
  A)
                  \[x\sin x\] done clear
  B)
                  \[x\cos x+\sin x+2\] done clear
  C)
                  \[x\sin x+2\] done clear
  D)
                  \[x\cos x+2\] done clear
  View Solution play_arrow

• question_answer38)
  \[\int{{{\cos }^{-1}}\left( \frac{1}{x} \right)\,\,dx}\]         [RPET 2002]
  A)
                  \[x{{\sec }^{-1}}x+{{\cosh }^{-1}}x+C\] done clear
  B)
                  \[x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+C\] done clear
  C)
                  \[x{{\sec }^{-1}}x-{{\sin }^{-1}}x+C\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer39)
  \[\int{{{x}^{3}}\log x\,\,dx=}\]   [Karnataka CET 2002]
  A)
                  \[\frac{{{x}^{4}}\log x}{4}+c\] done clear
  B)
                  \[\frac{1}{16}[4{{x}^{4}}\log x-{{x}^{4}}]+c\] done clear
  C)
                  \[\frac{1}{8}[{{x}^{4}}\log x-4{{x}^{2}}]+c\] done clear
  D)
   \[\frac{1}{16}[4{{x}^{4}}\log x+{{x}^{4}}]+c\] done clear
  View Solution play_arrow

• question_answer40)
  \[\int{\cos ({{\log }_{e}}x)\,dx}\] is equal to        [MP PET 2003]
  A)
                  \[\frac{1}{2}x\{\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)\}\] done clear
  B)
                  \[x\{\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)\}\] done clear
  C)
                  \[\frac{1}{2}x\{\cos ({{\log }_{e}}x)-\sin ({{\log }_{e}}x)\}\] done clear
  D)
                  \[x\{\cos ({{\log }_{e}}x)-\sin ({{\log }_{e}}x)\}\] done clear
  View Solution play_arrow

• question_answer41)
  \[\int_{{}}^{{}}{{{e}^{x}}(1+\tan x)\sec x\ dx=}\]               [Karnataka CET 2005]
  A)
                  \[{{e}^{x}}\cot x\] done clear
  B)
                  \[{{e}^{x}}\tan x\] done clear
  C)
                  \[{{e}^{x}}\sec x\] done clear
  D)
                  \[{{e}^{x}}\cos x\] done clear
  View Solution play_arrow

• question_answer42)
  \[\int_{{}}^{{}}{\frac{x{{e}^{x}}}{{{(1+x)}^{2}}}dx=}\] [MP PET 1997; UPSEAT 2001; RPET 2002]
  A)
                  \[\frac{{{e}^{-x}}}{1+x}+c\] done clear
  B)
                  \[-\frac{{{e}^{-x}}}{1+x}+c\] done clear
  C)
                  \[\frac{{{e}^{x}}}{1+x}+c\] done clear
  D)
                  \[-\frac{{{e}^{x}}}{1+x}+c\] done clear
  View Solution play_arrow

• question_answer43)
  \[\int_{{}}^{{}}{{{e}^{x}}[\tan x-\log (\cos x)]\ dx=}\]      [MP PET 1991]
  A)
                  \[{{e}^{x}}\log (\sec x)+c\] done clear
  B)
                  \[{{e}^{x}}\log (\cos \text{ec}x)+c\] done clear
  C)
                  \[{{e}^{x}}\log (\cos x)+c\] done clear
  D)
                  \[{{e}^{x}}\log (\sin x)+c\] done clear
  View Solution play_arrow

• question_answer44)
  \[\int_{{}}^{{}}{{{e}^{x}}\sin x(\sin x+2\cos x)}\ dx=\]            [MP PET 1988]
  A)
                  \[{{e}^{x}}{{\sin }^{2}}x+c\] done clear
  B)
                  \[{{e}^{x}}\sin x+c\] done clear
  C)
                  \[{{e}^{x}}\sin 2x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer45)
  \[\int_{{}}^{{}}{{{e}^{2x}}\frac{1+\sin 2x}{1+\cos 2x}}\ dx=\]
  A)
                  \[{{e}^{2x}}\tan x+c\] done clear
  B)
                  \[{{e}^{2x}}\cot x+c\] done clear
  C)
                  \[\frac{{{e}^{2x}}\tan x}{2}+c\] done clear
  D)
                  \[\frac{{{e}^{2x}}\cot x}{2}+c\] done clear
  View Solution play_arrow

• question_answer46)
  \[\int_{{}}^{{}}{\frac{{{e}^{x}}(x-1)}{{{x}^{2}}}\ dx=}\]
  A)
                  \[\frac{1}{x}{{e}^{x}}+c\] done clear
  B)
                  \[x{{e}^{-x}}+c\] done clear
  C)
                  \[\frac{1}{{{x}^{2}}}{{e}^{x}}+c\] done clear
  D)
                  \[\left( x-\frac{1}{x} \right){{e}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer47)
  \[\int_{{}}^{{}}{{{e}^{x}}\left[ \frac{1+x\log x}{x} \right]\ dx=}\]
  A)
                  \[{{e}^{x}}+\log x+c\] done clear
  B)
                  \[\frac{{{e}^{x}}}{\log x}+c\] done clear
  C)
                  \[{{e}^{x}}-\log x+c\] done clear
  D)
                  \[{{e}^{x}}\log x+c\] done clear
  View Solution play_arrow

• question_answer48)
  \[\int_{{}}^{{}}{{{e}^{x}}\left[ {{\sin }^{-1}}\frac{x}{a}+\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right]dx=}\]
  A)
                  \[\frac{1}{a}{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] done clear
  B)
                  \[a{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] done clear
  C)
                  \[{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] done clear
  D)
                  \[\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}+c\] done clear
  View Solution play_arrow

• question_answer49)
  \[\int_{{}}^{{}}{{{e}^{x}}\frac{({{x}^{2}}+1)}{{{(x+1)}^{2}}}dx=}\]
  A)
                  \[\left( \frac{x-1}{x+1} \right){{e}^{x}}+c\] done clear
  B)
                  \[{{e}^{x}}\left( \frac{x+1}{x-1} \right)+c\] done clear
  C)
                  \[{{e}^{x}}(x+1)(x-1)+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer50)
  \[\int_{{}}^{{}}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)}\,dx=\]          [AISSE 1983; MP PET 1994, 96]
  A)
                  \[-\frac{{{e}^{x}}}{{{x}^{2}}}+c\] done clear
  B)
                  \[\frac{{{e}^{x}}}{{{x}^{2}}}+c\] done clear
  C)
                  \[\frac{{{e}^{x}}}{x}+c\] done clear
  D)
                  \[-\frac{{{e}^{x}}}{x}+c\] done clear
  View Solution play_arrow

• question_answer51)
  \[\int{{{e}^{x}}(1+\tan x+{{\tan }^{2}}x)\,\,dx=}\]            [Karnataka CET 1999]
  A)
                  \[{{e}^{x}}\sin x+c\] done clear
  B)
                  \[{{e}^{x}}\cos x+c\] done clear
  C)
                  \[{{e}^{x}}\tan x+c\] done clear
  D)
    \[{{e}^{x}}\sec x+c\] done clear
  View Solution play_arrow

• question_answer52)
   is equal to          [RPET 2000]
  A)
                  \[-{{e}^{x}}\tan \,\left( x/2 \right)\] done clear
  B)
                  \[-{{e}^{x}}\cot \,\left( x/2 \right)\] done clear
  C)
                  \[-\frac{1}{2}{{e}^{x}}\tan \,\left( \frac{x}{2} \right)\] done clear
  D)
   \[\frac{1}{2}{{e}^{x}}\cot \,\left( \frac{x}{2} \right)\] done clear
  View Solution play_arrow

• question_answer53)
  \[\int{\frac{(x+3){{e}^{x}}}{{{(x+4)}^{2}}}\,\,dx=\,\,}\]   [Karnataka CET 2000]
  A)
                  \[\frac{1}{{{(x+4)}^{2}}}+c\] done clear
  B)
                  \[\frac{{{e}^{x}}}{{{(x+4)}^{2}}}+c\] done clear
  C)
                  \[\frac{{{e}^{x}}}{x+4}+c\] done clear
  D)
    \[\frac{{{e}^{x}}}{x+3}+c\] done clear
  View Solution play_arrow

• question_answer54)
  \[\int{{{\left( \frac{x+2}{x+4} \right)}^{2}}{{e}^{x}}\,\,dx}\] is equal to   [AMU 2000]
  A)
                  \[{{e}^{x}}\left( \frac{x}{x+4} \right)+c\] done clear
  B)
                  \[{{e}^{x}}\left( \frac{x+2}{x+4} \right)+c\] done clear
  C)
                  \[{{e}^{x}}\left( \frac{x-2}{x+4} \right)+c\] done clear
  D)
    \[\left( \frac{2x{{e}^{x}}}{x+4} \right)+c\] done clear
  View Solution play_arrow

• question_answer55)
  \[\int{{{e}^{x}}\left[ f(x)+{f}'(x) \right]\,\,dx}\] is equal to            [DCE 2002]
  A)
                  \[{{e}^{x}}f(x)\] done clear
  B)
                  \[{{e}^{x}}\] done clear
  C)
                  \[{{e}^{x}}{f}'(x)\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer56)
  \[\int{{{e}^{x}}(1-\cot x+{{\cot }^{2}}x)\,\,dx}\] equals [MP PET 2002]
  A)
                  \[{{e}^{x}}\cot x+c\] done clear
  B)
                  \[{{e}^{x}}\text{cosec }x+c\] done clear
  C)
                  \[-{{e}^{x}}\cot x+c\] done clear
  D)
    \[-{{e}^{x}}\text{cosec}\,x+c\] done clear
  View Solution play_arrow

• question_answer57)
  \[\int_{{}}^{{}}{{{\sin }^{-1}}x\ dx}\]is equal to   [MP PET 2004]
  A)
                  \[\frac{1}{\sqrt{1-{{x}^{2}}}}+c\] done clear
  B)
                  \[x{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] done clear
  C)
                  \[{{\cos }^{-1}}x+c\] done clear
  D)
                  \[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer58)
  \[\int_{{}}^{{}}{\frac{x-\sin x}{1-\cos x}dx=}\]        [AISSE 1989]
  A)
                  \[x\cot \frac{x}{2}+c\] done clear
  B)
                  \[-x\cot \frac{x}{2}+c\] done clear
  C)
                  \[\cot \frac{x}{2}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer59)
  \[\int_{{}}^{{}}{\frac{x+\sin x}{1+\cos x}\ dx}\] is equal to            [Roorkee 1980; UPSEAT 1999]
  A)
                  ?\[x\tan \frac{x}{2}+c\] done clear
  B)
                  \[x\tan \ \frac{x}{2}+c\] done clear
  C)
                  \[x\tan x+c\] done clear
  D)
                  \[\frac{1}{2}x\tan x+c\] done clear
  View Solution play_arrow

• question_answer60)
  If \[\int_{{}}^{{}}{\frac{{{e}^{x}}(1+\sin x)dx}{1+\cos x}={{e}^{x}}f(x)+c}\], then \[f(x)=\]             [RPET 1997; Karnataka CET 2003, 05; Orissa JEE 2004]
  A)
                  \[\sin \frac{x}{2}\] done clear
  B)
                  \[\cos \frac{x}{2}\] done clear
  C)
                  \[\tan \frac{x}{2}\] done clear
  D)
                  \[\log \frac{x}{2}\] done clear
  View Solution play_arrow

• question_answer61)
  \[\int_{{}}^{{}}{\sqrt{x}{{e}^{\sqrt{x}}}\ dx=}\]  [Karnataka CET 2004]
  A)
                  \[2\sqrt{x}-{{e}^{\sqrt{x}}}-4\sqrt{x}\ {{e}^{\sqrt{x}}}+c\] done clear
  B)
                  \[(2x-4\sqrt{x}+4){{e}^{\sqrt{x}}}+c\] done clear
  C)
                  \[(2x+4\sqrt{x}+4){{e}^{\sqrt{x}}}+c\] done clear
  D)
                  \[(1-4\sqrt{x}){{e}^{\sqrt{x}}}+c\] done clear
  View Solution play_arrow

• question_answer62)
  \[\int_{{}}^{{}}{32{{x}^{3}}{{(\log x)}^{2}}dx}\] is equal to             [MP PET 2004]
  A)
                  \[{{x}^{4}}\{8{{(\log x)}^{2}}-4(\log x)+1\}+c\]    done clear
  B)
                  \[{{x}^{3}}\{{{(\log x)}^{2}}+2\log x\}+c\] done clear
  C)
                  \[{{x}^{4}}\{8{{(\log x)}^{2}}-4\log x\}+c\]            done clear
  D)
                  \[8{{x}^{4}}{{(\log x)}^{2}}+c\] done clear
  View Solution play_arrow

• question_answer63)
  \[\int_{{}}^{{}}{{{\sin }^{-1}}(3x-4{{x}^{3}})dx=}\]             [AISSE 1986; DSSE 1984]
  A)
                  \[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] done clear
  B)
                  \[x{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] done clear
  C)
                  \[2[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\] done clear
  D)
                  \[3[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\] done clear
  View Solution play_arrow

• question_answer64)
  \[\int_{{}}^{{}}{\cos \sqrt{x}\ dx=}\]       [BIT Ranchi 1990; IIT 1977; RPET 1999]
  A)
                  \[2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\]          done clear
  B)
                  \[2[\sqrt{x}\sin \sqrt{x}-\cos \sqrt{x}]+c\] done clear
  C)
                  \[2[\cos \sqrt{x}-\sqrt{x}\sin \sqrt{x}]+c\]           done clear
  D)
                  \[-2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\] done clear
  View Solution play_arrow

• question_answer65)
  \[\int_{{}}^{{}}{{{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}}dx=}\]       [MP PET 1991]
  A)
                  \[x{{\tan }^{-1}}x+c\]     done clear
  B)
                  \[x{{\tan }^{-1}}x-\log (1+{{x}^{2}})+c\] done clear
  C)
                  \[2x{{\tan }^{-1}}x+\log (1+{{x}^{2}})+c\]             done clear
  D)
                  \[2x{{\tan }^{-1}}x-\log (1+{{x}^{2}})+c\] done clear
  View Solution play_arrow

• question_answer66)
  \[\int_{{}}^{{}}{\frac{x{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\ }dx=\]           [MNR 1978; EAMCET 1982; IIT 1984]
  A)
                  \[x-\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+c\] done clear
  B)
                  \[x+\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+c\] done clear
  C)
                  \[\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x-x+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\int_{{}}^{{}}{\frac{{{\sin }^{-1}}x}{{{(1-{{x}^{2}})}^{3/2}}}\ dx=}\]     [AISSE 1983, 87]
  A)
                  \[\frac{x}{\sqrt{1-{{x}^{2}}}}{{\sin }^{-1}}x+\frac{1}{2}\log (1-{{x}^{2}})+c\] done clear
  B)
                  \[\frac{x}{\sqrt{1-{{x}^{2}}}}{{\sin }^{-1}}x-\frac{1}{2}\log (1-{{x}^{2}})+c\] done clear
  C)
                  \[\frac{1}{\sqrt{1-{{x}^{2}}}}{{\sin }^{-1}}x-\frac{1}{2}\log (1-{{x}^{2}})+c\] done clear
  D)
                  \[\frac{1}{\sqrt{1-{{x}^{2}}}}{{\sin }^{-1}}x+\frac{1}{2}\log (1-{{x}^{2}})+c\] done clear
  View Solution play_arrow

• question_answer68)
  \[\int_{{}}^{{}}{\frac{x{{\tan }^{-1}}x}{{{(1+{{x}^{2}})}^{3/2}}}\ dx=}\]
  A)
                  \[\frac{x+{{\tan }^{-1}}x}{\sqrt{1+{{x}^{2}}}}+c\] done clear
  B)
                  \[\frac{x-{{\tan }^{-1}}x}{\sqrt{1+{{x}^{2}}}}+c\] done clear
  C)
                  \[\frac{{{\tan }^{-1}}x-x}{\sqrt{1+{{x}^{2}}}}+c\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer69)
  \[\int_{{}}^{{}}{{{x}^{5}}.{{e}^{{{x}^{2}}}}dx=}\]
  A)
                  \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\]      done clear
  B)
                  \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}+{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\] done clear
  C)
                  \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}-{{e}^{{{x}^{2}}}}+c\]       done clear
  D)
                  None of these done clear
  View Solution play_arrow

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

• question_answer71)
  \[\int_{{}}^{{}}{{{e}^{\sqrt{x}}}\ dx}\] is equal to               [MP PET 1998]
  A)
                  \[{{e}^{\sqrt{x}}}+A\] done clear
  B)
                  \[\frac{1}{2}{{e}^{\sqrt{x}}}+A\] done clear
  C)
                  \[2(\sqrt{x}-1){{e}^{\sqrt{x}}}+A\] done clear
  D)
                  \[2(\sqrt{x}+1){{e}^{\sqrt{x}}}+A\] (A is an arbitrary constant) done clear
  View Solution play_arrow

• question_answer72)
  \[{{I}_{1}}=\int{{{\sin }^{-1}}x\,\,dx}\] and \[{{I}_{2}}=\int{{{\sin }^{-1}}\sqrt{1-{{x}^{2}}}}dx\]then                 [Kerala (Engg.) 2005]
  A)
                  \[{{I}_{1}}={{I}_{2}}\] done clear
  B)
                  \[{{I}_{2}}=\pi /2{{I}_{1}}\] done clear
  C)
                  \[{{I}_{1}}+{{I}_{2}}=\pi /2x\] done clear
  D)
                  \[{{I}_{1}}+{{I}_{2}}=\pi /2\] done clear
  E)
                  \[{{I}_{1}}-{{I}_{2}}=\pi /2x\] done clear
  View Solution play_arrow

• question_answer73)
  If an ant derivative of \[f(x)\] is \[{{e}^{x}}\] and that of \[g(x)\] is \[\cos x,\]then \[\int{f(x)\cos x\,dx}+\int{g(x){{e}^{x}}dx=}\] [Kerala (Engg.) 2005]
  A)
                  f(x)g(x)+c done clear
  B)
                  f(x)+g(x)+c done clear
  C)
                  \[{{e}^{x}}\cos x+c\] done clear
  D)
                  f(x) ? g(x)+c done clear
  E)
                  \[{{e}^{x}}\cos x+f(x)g(x)+c\] done clear
  View Solution play_arrow