JEE Main & Advanced Mathematics Indefinite Integrals Question Bank

done Fundamental Integration Total Questions - 87

• question_answer1)
  \[\int_{{}}^{{}}{\sqrt{1-\sin 2x}\ }dx=........,\ \ x\in (0,\ \pi /4)\] [MP PET 1987]
  A)
             \[-\sin x+\cos x\] done clear
  B)
             \[\sin x-\cos x\] done clear
  C)
             \[\tan x+\sec x\] done clear
  D)
             \[\sin x+\cos x\] done clear
  View Solution play_arrow

• question_answer2)
  \[\int_{{}}^{{}}{\frac{1+{{\cos }^{2}}x}{{{\sin }^{2}}x}dx}=\]  [MP PET 1993; BIT Ranchi 1982]
  A)
             \[-\cot x-2x+c\] done clear
  B)
             \[-2\cot x-2x+c\] done clear
  C)
             \[-2\cot x-x+c\] done clear
  D)
             \[-2\cot x+x+c\] done clear
  View Solution play_arrow

• question_answer3)
  \[\int_{{}}^{{}}{{{\sin }^{-1}}}(\cos x)dx=\]
  A)
             \[\frac{\pi x}{2}\] done clear
  B)
             \[\frac{\pi {{x}^{2}}}{2}\] done clear
  C)
             \[\frac{\pi x-{{x}^{2}}}{2}\] done clear
  D)
             \[\frac{\pi x+{{x}^{2}}}{2}\] done clear
  View Solution play_arrow

• question_answer4)
  \[\int_{{}}^{{}}{\frac{dx}{\tan x+\cot x}}=\] [MP PET 1991]
  A)
             \[\frac{\cos 2x}{4}+c\] done clear
  B)
             \[\frac{\sin 2x}{4}+c\] done clear
  C)
             \[-\frac{\sin 2x}{4}+c\] done clear
  D)
             \[-\frac{\cos 2x}{4}+c\] done clear
  View Solution play_arrow

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

• question_answer6)
  If \[f'(x)={{x}^{2}}+5\] and \[f(0)=-1\], then \[f(x)=\]
  A)
             \[{{x}^{3}}+5x-1\] done clear
  B)
             \[{{x}^{3}}+5x+1\] done clear
  C)
             \[\frac{1}{3}{{x}^{3}}+5x-1\] done clear
  D)
             \[\frac{1}{3}{{x}^{3}}+5x+1\] done clear
  View Solution play_arrow

• question_answer7)
  \[\int_{{}}^{{}}{{{\tan }^{-1}}\sqrt{\frac{1-\cos 2x}{1+\cos 2x}}}\ dx=\]
  A)
             \[2{{x}^{2}}+c\] done clear
  B)
             \[{{x}^{2}}+c\] done clear
  C)
             \[\frac{{{x}^{2}}}{2}+c\] done clear
  D)
             \[2x+c\] done clear
  View Solution play_arrow

• question_answer8)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x}+\sqrt{x-2}}=}\] [MP PET 1990]
  A)
             \[\frac{1}{3}[{{x}^{3/2}}-{{(x-2)}^{3/2}}]+c\] done clear
  B)
             \[\frac{2}{3}[{{x}^{3/2}}-{{(x-2)}^{3/2}}]+c\] done clear
  C)
             \[\frac{1}{3}[{{(x-2)}^{3/2}}-{{x}^{3/2}}]+c\]                               done clear
  D)
             \[\frac{2}{3}[{{(x-2)}^{3/2}}-{{x}^{3/2}}]+c\] done clear
  View Solution play_arrow

• question_answer9)
  \[\int_{{}}^{{}}{\frac{\sin x}{\sin (x-\alpha )}dx=}\] [RPET 1999; Kerala (Engg.) 2002; AIEEE 2004]
  A)
             \[x\cos \alpha -\sin \alpha \log \sin (x-\alpha )+c\] done clear
  B)
             \[x\cos \alpha +\sin \alpha \log \sin (x-\alpha )+c\] done clear
  C)
             \[x\sin \alpha -\sin \alpha \log \sin (x-\alpha )+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer10)
  \[\int_{{}}^{{}}{\frac{\cos x-1}{\cos x+1}\ dx=}\]       [MP PET 1989, 92]
  A)
             \[2\tan \frac{x}{2}-x+c\] done clear
  B)
             \[\frac{1}{2}\tan \frac{x}{2}-x+c\] done clear
  C)
             \[x-\frac{1}{2}\tan \frac{x}{2}+c\] done clear
  D)
             \[x-2\tan \frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer11)
  \[\int_{{}}^{{}}{\frac{dx}{1-\sin x}}=\]                              [MP PET 1991]
  A)
             \[x+\cos x+c\] done clear
  B)
             \[1+\sin x+c\] done clear
  C)
             \[\sec x-\tan x+c\] done clear
  D)
             \[\sec x+\tan x+c\] done clear
  View Solution play_arrow

• question_answer12)
  If \[\int_{{}}^{{}}{(\sin 2x-\cos 2x)}\ dx=\frac{1}{\sqrt{2}}\sin (2x-a)+b\], then [Roorkee 1978; MP PET 2001]
  A)
             \[a=\frac{\pi }{4},\ b=0\] done clear
  B)
             \[a=-\frac{\pi }{4},\ b=0\] done clear
  C)
             \[a=\frac{5\pi }{4},\ b=\]any constant                                         done clear
  D)
             \[a=-\frac{5\pi }{4},\ b=\]any constant done clear
  View Solution play_arrow

• question_answer13)
  \[\int_{{}}^{{}}{\left( 1+x+\frac{{{x}^{2}}}{2\ !}+\frac{{{x}^{3}}}{3\ !}+.......... \right)\ dx=}\]
  A)
             \[-{{e}^{x}}+c\] done clear
  B)
             \[{{e}^{x}}+c\] done clear
  C)
             \[{{e}^{-x}}+c\] done clear
  D)
             \[-{{e}^{-x}}+c\] done clear
  View Solution play_arrow

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

• question_answer15)
  \[\int_{{}}^{{}}{{{(\sec x+\tan x)}^{2}}dx=}\]              [MP PET 1987, 92]
  A)
             \[2(\sec x+\tan x)-x+c\] done clear
  B)
             \[1/3{{(\sec x+\tan x)}^{3}}+c\] done clear
  C)
             \[\sec x(\sec x+\tan x)+c\] done clear
  D)
             \[2(\sec x+\tan x)+c\] done clear
  View Solution play_arrow

• question_answer16)
  \[\int_{{}}^{{}}{{{x}^{51}}({{\tan }^{-1}}x+{{\cot }^{-1}}x)\ dx=}\] [MP PET 1991]
  A)
             \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\] done clear
  B)
             \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\] done clear
  C)
             \[\frac{\pi {{x}^{52}}}{104}+\frac{\pi }{2}+c\] done clear
  D)
             \[\frac{{{x}^{52}}}{52}+\frac{\pi }{2}+c\] done clear
  View Solution play_arrow

• question_answer17)
  \[\int_{{}}^{{}}{5\sin xdx=}\] [MP PET 1988]
  A)
             \[5\cos x+c\] done clear
  B)
             \[-5\cos x+c\] done clear
  C)
             \[5\sin x+c\] done clear
  D)
             \[-5\sin x+c\] done clear
  View Solution play_arrow

• question_answer18)
  \[\int_{{}}^{{}}{\frac{\tan x}{\sec x+\tan x}\ dx=}\]
  A)
             \[\sec x+\tan x-x+c\] done clear
  B)
             \[\sec x-\tan x+x+c\] done clear
  C)
             \[\sec x+\tan x+x+c\] done clear
  D)
             \[-\sec x-\tan x+x+c\] done clear
  View Solution play_arrow

• question_answer19)
  \[\int_{{}}^{{}}{\frac{dx}{{{\sin }^{2}}x{{\cos }^{2}}x}=}\]  [Roorkee 1976; RPET 1991]
  A)
             \[\tan x+\cot x+c\] done clear
  B)
             \[\cot x-\tan x+c\] done clear
  C)
             \[\tan x-\cot x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

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

• question_answer21)
  \[\int_{{}}^{{}}{\sqrt{1+\sin \frac{x}{2}}\ dx=}\] [IIT 1980; MP PET 1989; Pb. CET 2003]
  A)
             \[\frac{1}{4}\left( \cos \frac{x}{4}-\sin \frac{x}{4} \right)+c\] done clear
  B)
             \[4\left( \cos \frac{x}{4}-\sin \frac{x}{4} \right)+c\] done clear
  C)
             \[4\left( \sin \frac{x}{4}-\cos \frac{x}{4} \right)+c\] done clear
  D)
             \[4\left( \sin \frac{x}{4}+\cos \frac{x}{4} \right)+c\] done clear
  View Solution play_arrow

• question_answer22)
  \[\int_{{}}^{{}}{({{\sin }^{-1}}x+{{\cos }^{-1}}x)\ dx=}\] [MP PET 1990]
  A)
             \[\frac{1}{2}\pi x+c\] done clear
  B)
             \[x({{\sin }^{-1}}x-{{\cos }^{-1}}x)+c\] done clear
  C)
             \[x({{\cos }^{-1}}x+{{\sin }^{-1}}x)+c\] done clear
  D)
             \[\frac{\pi }{2}+x+c\] done clear
  View Solution play_arrow

• question_answer23)
  \[\int_{{}}^{{}}{\frac{\sin x+\cos x}{\sqrt{1+\sin 2x}}\ dx=}\] [MP PET 1990]
  A)
             \[\sin x+c\] done clear
  B)
             \[\cos x+c\] done clear
  C)
             \[x+c\] done clear
  D)
             \[{{x}^{2}}+c\] done clear
  View Solution play_arrow

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

• question_answer25)
  \[\int_{{}}^{{}}{\frac{\text{cosec}\theta -\cot \theta }{\text{cosec}\theta +\cot \theta }}\ d\theta =\]
  A)
             \[2\text{cosec}\theta -2\cot \theta -\theta +c\]                         done clear
  B)
             \[2\,\text{cosec}\theta -2\cot \theta +\theta +c\] done clear
  C)
             \[2\,\text{cosec}\theta +2\cot \theta -\theta +c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer26)
  \[\int_{{}}^{{}}{(1+2x+3{{x}^{2}}+4{{x}^{3}}+......)\ dx=}\]
  A)
             \[{{(1+x)}^{-1}}+c\] done clear
  B)
             \[{{(1-x)}^{-1}}+c\] done clear
  C)
             \[{{(1-x)}^{-1}}-1+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer27)
  If \[\int_{{}}^{{}}{(\cos x-\sin x)\ dx=\sqrt{2}\sin (x+\alpha )+c}\], then \[\alpha =\]  
  A)
             \[\frac{\pi }{3}\] done clear
  B)
             \[-\frac{\pi }{3}\] done clear
  C)
             \[\frac{\pi }{4}\] done clear
  D)
             \[-\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer28)
  \[\int_{{}}^{{}}{\frac{3{{x}^{2}}-2\sqrt{x}}{x}}dx=\] [Roorkee 1976]
  A)
             \[{{x}^{3}}-\sqrt{x}+c\] done clear
  B)
             \[{{x}^{3}}+\sqrt{x}+c\] done clear
  C)
             \[{{x}^{3}}-2\sqrt{x}+c\] done clear
  D)
             \[{{x}^{3}}-4\sqrt{x}+c\] done clear
  View Solution play_arrow

• question_answer29)
  If \[\int_{{}}^{{}}{\frac{dx}{1+\sin x}=\tan \left( \frac{x}{2}+a \right)+b}\], then [Roorkee 1979]
  A)
             \[a=\frac{\pi }{4},\ b=3\]          done clear
  B)
             \[a=-\frac{\pi }{4},\ b=3\] done clear
  C)
             \[a=\frac{\pi }{4},\ b=\]arbitrary constant                                   done clear
  D)
             \[a=-\frac{\pi }{4},\ b=\]arbitrary constant done clear
  View Solution play_arrow

• question_answer30)
  \[\int_{{}}^{{}}{\frac{dx}{\sin x+\cos x}}=\] [BIT Ranchi 1990; RPET 1997; Karnataka CET 1999; Orissa JEE 2004]
  A)
             \[\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\] done clear
  B)
             \[\log \tan \left( \frac{\pi }{8}-\frac{x}{2} \right)+c\] done clear
  C)
             \[\frac{1}{\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer31)
  \[\int_{{}}^{{}}{\frac{1}{x\sqrt{{{x}^{2}}-1}}\ dx=}\] [MP PET 1988]
  A)
             \[{{\cos }^{-1}}x+c\] done clear
  B)
             \[{{\sec }^{-1}}x+c\] done clear
  C)
             \[{{\cot }^{-1}}x+c\] done clear
  D)
             \[{{\tan }^{-1}}x+c\] done clear
  View Solution play_arrow

• question_answer32)
  \[\int_{{}}^{{}}{\frac{\cos 2x+2{{\sin }^{2}}x}{{{\cos }^{2}}x}dx=}\]
  A)
             \[2\sec x+c\] done clear
  B)
             \[2\tan x+c\] done clear
  C)
             \[\tan x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer33)
  \[\int_{{}}^{{}}{\frac{{{\sin }^{3}}x+{{\cos }^{3}}x}{{{\sin }^{2}}x{{\cos }^{2}}x}}\ dx=\]
  A)
             \[\tan x+\cot x+c\] done clear
  B)
             \[\tan x-\cot x+c\] done clear
  C)
             \[\text{cosec}\,x-\cot x+c\] done clear
  D)
             \[\sec x-\text{cosec}\,x+c\] done clear
  View Solution play_arrow

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

• question_answer35)
  \[\int_{{}}^{{}}{\frac{1}{{{x}^{2}}}{{(2x+1)}^{3}}}dx=\]
  A)
             \[4{{x}^{2}}+12x+6\log x-\frac{1}{x}+c\]                                      done clear
  B)
             \[4{{x}^{2}}+12x-6\log x-\frac{2}{x}+c\] done clear
  C)
             \[2{{x}^{2}}+8x+3\log x-\frac{2}{x}+c\]                                        done clear
  D)
             \[8{{x}^{2}}+6x+6\log x+\frac{2}{x}+c\] done clear
  View Solution play_arrow

• question_answer36)
  \[\int_{{}}^{{}}{\frac{5({{x}^{6}}+1)}{{{x}^{2}}+1}dx=}\]
  A)
             \[5({{x}^{7}}+x){{\tan }^{-1}}x+c\] done clear
  B)
             \[{{x}^{5}}-\frac{5}{3}{{x}^{3}}+5x+c\] done clear
  C)
             \[3{{x}^{4}}-5{{x}^{2}}+15x+c\] done clear
  D)
             \[5{{\tan }^{-1}}({{x}^{2}}+1)+\log ({{x}^{2}}+1)+c\] done clear
  View Solution play_arrow

• question_answer37)
  \[\int_{{}}^{{}}{\frac{a{{x}^{-2}}+b{{x}^{-1}}+c}{{{x}^{-3}}}}\ dx=\]
  A)
             \[2a{{x}^{2}}+3b{{x}^{3}}+4c{{x}^{4}}+k\]                                    done clear
  B)
             \[6a{{x}^{2}}+4b{{x}^{3}}+3c{{x}^{4}}+k\] done clear
  C)
             \[a+b+c{{x}^{2}}+k\]                  done clear
  D)
             \[\frac{1}{2}a{{x}^{2}}+\frac{1}{3}b{{x}^{3}}+\frac{1}{4}c{{x}^{4}}+k\] done clear
  View Solution play_arrow

• question_answer38)
  \[\int_{{}}^{{}}{\frac{dx}{\sin x+\sqrt{3}\cos x}}=\]
  A)
             \[\log \tan \left( \frac{x}{2}+\frac{\pi }{2} \right)+c\] done clear
  B)
             \[\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\] done clear
  C)
             \[\log \cot \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\] done clear
  D)
             \[\frac{1}{2}\log \cot \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\] done clear
  View Solution play_arrow

• question_answer39)
  \[\int_{{}}^{{}}{\frac{{{x}^{2}}+x-6}{(x-2)(x-1)}dx=}\]
  A)
             \[x+2\log (x-1)+c\] done clear
  B)
             \[2x+2\log (x-1)+c\] done clear
  C)
             \[x+4\log (1-x)+c\] done clear
  D)
             \[x+4\log (x-1)+c\] done clear
  View Solution play_arrow

• question_answer40)
  \[\int_{{}}^{{}}{\frac{dx}{4{{\cos }^{3}}2x-3\cos 2x}}=\]
  A)
             \[\frac{1}{3}\log [\sec 6x+\tan 6x]+c\] done clear
  B)
             \[\frac{1}{6}\log [\sec 6x+\tan 6x]+c\] done clear
  C)
             \[\log [\sec 6x+\tan 6x]+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer41)
  \[\int_{{}}^{{}}{\frac{\sin 3x}{\sin x}\ dx=}\]
  A)
             \[x+\sin 2x+c\] done clear
  B)
             \[3x+\sin 2x+c\] done clear
  C)
             \[3x+{{\sin }^{2}}x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer42)
  If \[\int_{{}}^{{}}{\frac{f(x)\ dx}{\log \sin x}=\log \log \sin x}\], then \[f(x)=\]
  A)
             \[\sin x\] done clear
  B)
             \[\cos x\] done clear
  C)
             \[\log \sin x\] done clear
  D)
             \[\cot x\] done clear
  View Solution play_arrow

• question_answer43)
  \[\int_{{}}^{{}}{\frac{\sin x+\text{cosec}\,x}{\tan x}dx=}\]
  A)
             \[\sin x-\text{cosec}\,x+c\] done clear
  B)
             \[\text{cosec}\,x-\sin x+c\] done clear
  C)
             \[\log \tan x+c\] done clear
  D)
             \[\log \cot x+c\] done clear
  View Solution play_arrow

• question_answer44)
  \[\int_{{}}^{{}}{\frac{1}{\sqrt{1+\sin x}}dx}=\] [RPET 1996]
  A)
             \[2\sqrt{2}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\] done clear
  B)
             \[\frac{1}{\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\] done clear
  C)
             \[\sqrt{2}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\] done clear
  D)
             \[\frac{1}{2\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\] done clear
  View Solution play_arrow

• question_answer45)
  \[\int_{{}}^{{}}{{{(\tan x-\cot x)}^{2}}\ dx=}\]
  A)
             \[\tan x+\cot x+c\] done clear
  B)
             \[\sec x\tan x+c\] done clear
  C)
             \[\cos \text{ec}x\cot x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer46)
  \[\int_{{}}^{{}}{{{\{1+2\tan x(\tan x+\sec x)\}}^{1/2}}dx=}\] [Roorkee 1987]
  A)
             \[\log (\sec x+\tan x)+c\]          done clear
  B)
             \[\log {{(\sec x+\tan x)}^{1/2}}+c\] done clear
  C)
             \[\log \sec x(\sec x+\tan x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer47)
  \[\int_{{}}^{{}}{\frac{2x}{{{(2x+1)}^{2}}}dx=}\] [DSSE 1985]
  A)
             \[\frac{1}{2}\log (2x+1)+\frac{1}{2(2x+1)}+c\]                            done clear
  B)
             \[\frac{1}{2}\log (2x+1)-\frac{1}{2(2x+1)}+c\] done clear
  C)
             \[2\log (2x+1)+\frac{1}{2(2x+1)}+c\]                                           done clear
  D)
             \[2\log (2x+1)-\frac{1}{2(2x+1)}+c\] done clear
  View Solution play_arrow

• question_answer48)
  \[\int_{{}}^{{}}{\frac{{{\sin }^{2}}x-{{\cos }^{2}}x}{{{\sin }^{2}}x{{\cos }^{2}}x}dx=}\] [MP PET 1996]
  A)
             \[\tan x+\cot x+c\] done clear
  B)
             \[\tan x+\text{cosec}\,x+c\] done clear
  C)
             \[-\tan x+\cot x+c\] done clear
  D)
             \[\tan x+\sec x+c\] done clear
  View Solution play_arrow

• question_answer49)
  \[\int_{{}}^{{}}{(3\,\text{cose}{{\text{c}}^{2}}x+2\sin 3x)\ dx=}\] [AI CBSE 1981]
  A)
             \[3\cot x+\frac{2}{3}\cos 3x+c\]       done clear
  B)
             \[-\left( 3\cot x+\frac{2}{3}\cos 3x \right)+c\] done clear
  C)
             \[3\cot x-\frac{2}{3}\cos 3x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer50)
  If \[f'(x)=\frac{1}{x}+x\] and \[f(1)=\frac{5}{2}\], then \[f(x)=\]
  A)
             \[\log x+\frac{{{x}^{2}}}{2}+2\] done clear
  B)
             \[\log x+\frac{{{x}^{2}}}{2}+1\] done clear
  C)
             \[\log x-\frac{{{x}^{2}}}{2}+2\] done clear
  D)
             \[\log x-\frac{{{x}^{2}}}{2}+1\] done clear
  View Solution play_arrow

• question_answer51)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{1+x}+\sqrt{x}}=}\]
  A)
             \[\frac{2}{3}{{(1+x)}^{2/3}}-\frac{2}{3}{{x}^{2/3}}+c\]                done clear
  B)
             \[\frac{3}{2}{{(1+x)}^{2/3}}+\frac{3}{2}{{x}^{2/3}}+c\] done clear
  C)
             \[\frac{3}{2}{{(1+x)}^{3/2}}+\frac{3}{2}{{x}^{3/2}}+c\]               done clear
  D)
             \[\frac{2}{3}{{(1+x)}^{3/2}}-\frac{2}{3}{{x}^{3/2}}+c\] done clear
  View Solution play_arrow

• question_answer52)
  \[\int_{{}}^{{}}{\frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }}dx=\] [MP PET 1994]
  A)
             \[2[\sin x+x\cos \alpha ]+c\] done clear
  B)
             \[2[\sin x+\sin \alpha ]+c\] done clear
  C)
             \[2[-\sin x+x\cos \alpha ]+c\] done clear
  D)
             \[-2[\sin x+\sin \alpha ]+c\] done clear
  View Solution play_arrow

• question_answer53)
  \[\int_{{}}^{{}}{\frac{1-\tan x}{1+\tan x}\ dx=}\] [MP PET 1994]
  A)
             \[\log \sec \left( \frac{\pi }{4}-x \right)+c\] done clear
  B)
             \[\log \cos \left( \frac{\pi }{4}+x \right)+c\] done clear
  C)
             \[\log \sin \left( \frac{\pi }{4}+x \right)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer54)
  \[\int_{{}}^{{}}{{{a}^{x}}\ da=}\] [MP PET 1994, 96]
  A)
             \[\frac{{{a}^{x}}}{{{\log }_{e}}a}+c\] done clear
  B)
             \[{{a}^{x}}{{\log }_{e}}a+c\]         done clear
  C)
             \[\frac{{{a}^{x+1}}}{x+1}+c\] done clear
  D)
             \[x{{a}^{x-1}}+c\] done clear
  View Solution play_arrow

• question_answer55)
  The value of \[\int_{{}}^{{}}{\cot x\ dx}\] is [RPET 1995]
  A)
             \[\log \cos x+c\] done clear
  B)
             \[\log \tan x+c\]           done clear
  C)
             \[\log \sin x+c\] done clear
  D)
             \[\log \sec x+c\] done clear
  View Solution play_arrow

• question_answer56)
  The value of \[\int_{{}}^{{}}{\frac{1}{{{x}^{4}}}\ dx}\] is [RPET 1995]
  A)
             \[\frac{1}{-3{{x}^{3}}}+c\] done clear
  B)
             \[\frac{1}{3{{x}^{3}}}+c\] done clear
  C)
             \[\frac{1}{-4{{x}^{3}}}+c\] done clear
  D)
             \[-\frac{1}{3{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer57)
  \[\int_{{}}^{{}}{\frac{{{e}^{5\log x}}-{{e}^{4\log x}}}{{{e}^{3\log x}}-{{e}^{2\log x}}}\ dx=}\] [MNR 1985]
  A)
             \[e\ .\ {{3}^{-3x}}+c\] done clear
  B)
             \[{{e}^{3}}\log x+c\] done clear
  C)
             \[\frac{{{x}^{3}}}{3}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer58)
  \[\int_{{}}^{{}}{\frac{{{x}^{4}}+{{x}^{2}}+1}{{{x}^{2}}-x+1}\ dx=}\]
  A)
             \[\frac{1}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}}+x+c\] done clear
  B)
             \[\frac{1}{3}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+x+c\] done clear
  C)
             \[\frac{1}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}}-x+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer59)
  \[\int_{{}}^{{}}{\sec x\ dx=}\] [MP PET 1988, 95; RPET 1996]
  A)
             \[\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\] done clear
  B)
             \[-\log (\sec x-\tan x)+c\] done clear
  C)
             \[\log (\sec x-\tan x)+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer60)
  \[\int_{{}}^{{}}{\sqrt{1+\sin x}\ dx=}\] [MP PET 1995]
  A)
             \[\frac{1}{2}\left( \sin \frac{x}{2}+\cos \frac{x}{2} \right)+c\] done clear
  B)
             \[\frac{1}{2}\left( \sin \frac{x}{2}-\cos \frac{x}{2} \right)+c\] done clear
  C)
             \[2\sqrt{1+\sin x}+c\] done clear
  D)
             \[-2\sqrt{1-\sin x}+c\] done clear
  View Solution play_arrow

• question_answer61)
  \[\int_{{}}^{{}}{\text{cose}{{\text{c}}^{2}}x\ dx}\] is equal to [MP PET 1999]
  A)
             \[\cot x+c\] done clear
  B)
             \[-\cot x+c\] done clear
  C)
             \[{{\tan }^{2}}x+c\] done clear
  D)
             \[-{{\cot }^{2}}x+c\] done clear
  View Solution play_arrow

• question_answer62)
  \[\int_{{}}^{{}}{\left( 2\sin x+\frac{1}{x} \right)\ dx}\] is equal to [MP PET 1999]
  A)
             \[-2\cos x+\log x+c\] done clear
  B)
             \[2\cos x+\log x+c\] done clear
  C)
             \[-2\sin x-\frac{1}{{{x}^{2}}}+c\] done clear
  D)
             \[-2\cos x+\frac{1}{{{x}^{2}}}+c\] done clear
  View Solution play_arrow

• question_answer63)
  \[\int_{{}}^{{}}{\sqrt{1+\cos x}\ dx}\] equals [RPET 1996]
  A)
             \[2\sqrt{2}\sin \frac{x}{2}+c\] done clear
  B)
             \[-2\sqrt{2}\sin \frac{x}{2}+c\] done clear
  C)
             \[-2\sqrt{2}\cos \frac{x}{2}+c\] done clear
  D)
             \[2\sqrt{2}\cos \frac{x}{2}+c\] done clear
  View Solution play_arrow

• question_answer64)
  \[\int_{{}}^{{}}{2\sin x}\cos x\ dx\]is equal to [SCRA 1996]
  A)
             \[\cos 2x+c\] done clear
  B)
             \[\sin 2x+c\] done clear
  C)
             \[{{\cos }^{2}}x+c\] done clear
  D)
             \[{{\sin }^{2}}x+c\] done clear
  View Solution play_arrow

• question_answer65)
  \[\int_{{}}^{{}}{{{\tan }^{2}}x\ dx}\] is equal to [SCRA 1996]
  A)
             \[\tan x+x+c\] done clear
  B)
             \[\tan x-x+c\] done clear
  C)
             \[\sec x+x+c\] done clear
  D)
             \[\sec x-x+c\] done clear
  View Solution play_arrow

• question_answer66)
  \[\int_{{}}^{{}}{{{e}^{\log (\sin x)}}dx=}\] [MP PET 1995]
  A)
             \[\sin x+c\] done clear
  B)
             \[-\cos x+c\] done clear
  C)
             \[{{e}^{\log (\cos x)}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\int_{{}}^{{}}{{{e}^{x\log a}}.\ {{e}^{x}}\ dx}\]is equal to [Kerala (Engg.) 2005]
  A)
             \[{{(ae)}^{x}}+c\] done clear
  B)
             \[\frac{{{(ae)}^{x}}}{\log (ae)}+c\] done clear
  C)
             \[\frac{{{e}^{x}}}{1+\log a}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer68)
  \[\int_{{}}^{{}}{\frac{1}{\sqrt{1+\cos x}}\ dx=}\]
  A)
             \[\sqrt{2}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\]             done clear
  B)
             \[\frac{1}{\sqrt{2}}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\] done clear
  C)
             \[\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\]            done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer69)
  \[\int{\frac{\cos 2x-1\,\,}{\cos 2x+1}dx=}\] [MP PET 2000]
  A)
             \[\tan x-x+c\] done clear
  B)
             \[x+\tan x+c\] done clear
  C)
             \[x-\tan x+c\] done clear
  D)
    \[-x-\cot x+c\] done clear
  View Solution play_arrow

• question_answer70)
  \[\int{\frac{a{{x}^{3}}+b{{x}^{2}}+c}{{{x}^{4}}}\,\,dx}\] equals to [RPET 2001]
  A)
             \[a\log x+\frac{b}{{{x}^{2}}}+\frac{c}{3{{x}^{3}}}+c\] done clear
  B)
             \[a\log x+\frac{b}{x}-\frac{c}{3{{x}^{3}}}+c\] done clear
  C)
             \[a\log x-\frac{b}{x}-\frac{c}{3{{x}^{3}}}+c\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer71)
  The value of \[\int{\frac{1}{{{(x-5)}^{2}}}\,\,dx}\] is [Karnataka CET 2001; Pb. CET 2002]
  A)
             \[\frac{1}{x-5}+c\] done clear
  B)
             \[-\frac{1}{x-5}+c\] done clear
  C)
             \[\frac{2}{{{\left( x-5 \right)}^{3}}}+c\] done clear
  D)
   \[-2{{\left( x-5 \right)}^{3}}+c\] done clear
  View Solution play_arrow

• question_answer72)
  If \[\int{f(x)\,dx=f(x)},\] then \[{{\int{\left[ f(x) \right]}}^{2}}\,\,dx\] is [DCE 2002]
  A)
             \[\frac{1}{2}{{\left[ f\left( x \right) \right]}^{2}}\] done clear
  B)
             \[{{\left[ f\left( x \right) \right]}^{3}}\] done clear
  C)
             \[\frac{{{\left[ f\left( x \right) \right]}^{3}}}{3}\] done clear
  D)
    \[{{\left[ \,f\left( x \right) \right]}^{2}}\] done clear
  View Solution play_arrow

• question_answer73)
  If \[\int{\sqrt{2}\sqrt{1+\sin x}}\,\,dx=-\,4\cos (ax+b)+c\] then the value of (a, b) is [UPSEAT 2002]
  A)
             \[\frac{1}{2},\,\frac{\pi }{4}\] done clear
  B)
             \[1,\,\frac{\pi }{2}\] done clear
  C)
             1, 1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer74)
  \[\int{{{13}^{x}}dx}\] is [Kerala (Engg.) 2002]
  A)
             \[\frac{{{13}^{x}}}{\log 13}+c\] done clear
  B)
             \[{{13}^{x+1}}+c\] done clear
  C)
             \[14x+c\] done clear
  D)
             \[{{14}^{x+1}}\]+ c done clear
  View Solution play_arrow

• question_answer75)
  \[\int{{{a}^{x}}\,\,dx=}\] [RPET 2003]
  A)
             \[\frac{{{a}^{x}}}{\log a}+c\] done clear
  B)
             \[{{a}^{x}}\log a+c\] done clear
  C)
             \[\log a+c\] done clear
  D)
             \[{{a}^{x}}+c\] done clear
  View Solution play_arrow

• question_answer76)
  \[\int{\sec x\tan x\,\,dx=}\] [RPET 2003]
  A)
             \[\sec x+\tan x+c\] done clear
  B)
             \[\sec x+c\] done clear
  C)
             \[\tan x+c\] done clear
  D)
    \[-\sec x+c\] done clear
  View Solution play_arrow

• question_answer77)
  \[\int{\left( {{\sin }^{4}}x-{{\cos }^{4}}x \right)\,dx=}\] [RPET 2003]
  A)
             \[-\frac{\cos 2x}{2}+c\] done clear
  B)
             \[-\frac{\sin 2x}{2}+c\] done clear
  C)
             \[\frac{\sin 2x}{2}+c\] done clear
  D)
    \[\frac{\cos 2x}{2}+c\] done clear
  View Solution play_arrow

• question_answer78)
  \[\int{\frac{{{(x+1)}^{2}}\,\,dx}{x({{x}^{2}}+1)}}\] is equal to  [MP PET 2003]
  A)
             \[{{\log }_{e}}x+c\] done clear
  B)
             \[{{\log }_{e}}x+2{{\tan }^{-1}}x+c\] done clear
  C)
             \[{{\log }_{e}}\frac{1}{{{x}^{2}}+1}+c\] done clear
  D)
    \[{{\log }_{e}}\{x({{x}^{2}}+1)\}+c\] done clear
  View Solution play_arrow

• question_answer79)
  The value of \[\int_{{}}^{{}}{\frac{dx}{\sqrt{1-x}}}\] is [Pb. CET 2001]
  A)
             \[2\sqrt{1-x}+c\] done clear
  B)
             \[-2\sqrt{1-x}+c\] done clear
  C)
             \[-{{\sin }^{-1}}\sqrt{x}+c\] done clear
  D)
              \[{{\sin }^{-1}}\sqrt{x}+c\] done clear
  View Solution play_arrow

• question_answer80)
  \[\int_{{}}^{{}}{\frac{dx}{1-{{x}^{2}}}=}\] [MP PET 1987, 92, 2000]
  A)
             \[{{\tan }^{-1}}x+c\] done clear
  B)
             \[{{\sin }^{-1}}x+c\] done clear
  C)
             \[\frac{1}{2}\ln \left| \frac{1+x}{1-x} \right|+c\] done clear
  D)
             \[\frac{1}{2}\ln \left| \frac{1-x}{1+x} \right|+c\] done clear
  View Solution play_arrow

• question_answer81)
  \[\int_{{}}^{{}}{\frac{dx}{4{{x}^{2}}+9}=}\] [MP PET 1991; Roorkee 1977; MNR 1974]
  A)
             \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] done clear
  B)
             \[\frac{3}{2}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] done clear
  C)
             \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{2x}{3} \right)+c\] done clear
  D)
             \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{3x}{2} \right)+c\] done clear
  View Solution play_arrow

• question_answer82)
  \[\int_{{}}^{{}}{\sqrt{1+{{x}^{2}}}\ dx=}\] [MP PET 1987, 89]
  A)
             \[\frac{x}{2}\sqrt{1+{{x}^{2}}}+\frac{1}{2}\log (x+\sqrt{1+{{x}^{2}}})+c\] done clear
  B)
             \[\frac{2}{3}{{(1+{{x}^{2}})}^{3/2}}+c\] done clear
  C)
             \[\frac{2}{3}x{{(1+{{x}^{2}})}^{3/2}}+c\]                                      done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer83)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}\] equals [SCRA 1996]
  A)
             \[{{\sin }^{-1}}\left( \frac{x}{a} \right)+c\] done clear
  B)
             \[{{\log }_{e}}|x+\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\] done clear
  C)
             \[{{\log }_{e}}|x-\sqrt{{{x}^{2}}-{{a}^{2}}}|+c\] done clear
  D)
             \[\frac{x\sqrt{{{x}^{2}}-{{a}^{2}}}}{2+c}\] done clear
  View Solution play_arrow

• question_answer84)
  \[\int{\frac{{{x}^{2}}}{{{x}^{2}}+4}\,\,dx}\] equals to [RPET 2001]
  A)
             \[x-2{{\tan }^{-1}}(x/2)+c\] done clear
  B)
             \[x+2{{\tan }^{-1}}(x/2)+c\] done clear
  C)
             \[x-4{{\tan }^{-1}}(x/2)+c\] done clear
  D)
    \[x+4{{\tan }^{-1}}(x/2)+c\] done clear
  View Solution play_arrow

• question_answer85)
  \[\int{\sqrt{{{x}^{2}}+{{a}^{2}}}\,\,dx}\] equals to                                                [RPET 2001]
  A)
             \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,-\frac{{{a}^{2}}}{2}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] done clear
  B)
             \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,+\frac{{{a}^{2}}}{2}\log \,\{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] done clear
  C)
             \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,-\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] done clear
  D)
             \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}\,+\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\] done clear
  View Solution play_arrow

• question_answer86)
  \[\int{\frac{dx}{{{a}^{2}}-{{x}^{2}}}}\] is equal to                                                  [EAMCET 2002]
  A)
             \[\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)\] done clear
  B)
             \[\frac{1}{2a}{{\sin }^{-1}}\left( \frac{a-x}{a+x} \right)\] done clear
  C)
             \[\frac{1}{2a}\log \,\left( \frac{a+x}{a-x} \right)\] done clear
  D)
   \[\frac{1}{2a}\log \,\left( \frac{a-x}{a+x} \right)\] done clear
  View Solution play_arrow

• question_answer87)
  \[\int_{{}}^{{}}{\frac{dx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}}\] is equal to                                [MP PET 2004]
  A)
             \[\frac{1}{2}x\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{1}{2}{{a}^{2}}\log (x+\sqrt{{{x}^{2}}+{{a}^{2}}})+c\]        done clear
  B)
             \[\frac{1}{2}\log ({{x}^{2}}+{{a}^{2}})+c\] done clear
  C)
             \[\log (x+\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] done clear
  D)
             \[\log (x-\sqrt{{{x}^{2}}+{{a}^{2}}})+c\] done clear
  View Solution play_arrow