-
question_answer1)
\[\int_{0}^{1}{{{e}^{2\,\text{In}\,x}}\,dx}=\] [MP PET 1990]
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{3}\] done
clear
D)
\[\frac{1}{4}\] done
clear
View Solution play_arrow
-
question_answer2)
\[\int_{0}^{\pi /4}{{{\tan }^{2}}x\,dx=}\] [Roorkee 1983, Pb. CET 2000]
A)
\[1-\frac{\pi }{4}\] done
clear
B)
\[1+\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{4}-1\] done
clear
D)
\[\frac{\pi }{4}\] done
clear
View Solution play_arrow
-
question_answer3)
\[\int_{0}^{\pi /2}{\frac{x+\sin x}{1+\cos x}\,dx=}\] [MP PET 1989]
A)
\[-\log 2\] done
clear
B)
\[\log 2\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer4)
\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,dx=}\] [Roorkee 1978]
A)
\[\frac{1}{2}({{e}^{\pi /2}}-1)\] done
clear
B)
\[\frac{1}{2}({{e}^{\pi /2}}+1)\] done
clear
C)
\[\frac{1}{2}(1-{{e}^{\pi /2}})\] done
clear
D)
\[2({{e}^{\pi /2}}+1)\] done
clear
View Solution play_arrow
-
question_answer5)
\[\int_{1}^{2}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)\,dx=}\] [MNR 1990; AMU 1999; UPSEAT 2000; Pb. CET 2004]
A)
\[\frac{{{e}^{2}}}{2}+e\] done
clear
B)
\[e-\frac{{{e}^{2}}}{2}\] done
clear
C)
\[\frac{{{e}^{2}}}{2}-e\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer6)
\[\int_{0}^{\pi /2}{\frac{\cos x}{(1+\sin x)(2+\sin x)}}\,dx=\] [UPSEAT 1999]
A)
\[\log \frac{4}{3}\] done
clear
B)
\[\log \frac{1}{3}\] done
clear
C)
\[\log \frac{3}{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer7)
\[\int_{\pi /3}^{\pi /2}{\frac{\sqrt{1+\cos x}}{{{(1-\cos x)}^{\frac{5}{2}}}}}\,dx=\] [AI CBSE 1980]
A)
\[\frac{5}{2}\] done
clear
B)
\[\frac{3}{2}\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
\[\frac{2}{5}\] done
clear
View Solution play_arrow
-
question_answer8)
\[\int_{1}^{2}{\frac{1}{{{x}^{2}}}{{e}^{\frac{-1}{x}}}\,dx=}\] [DCE 2001]
A)
\[\sqrt{e}+1\] done
clear
B)
\[\sqrt{e}-1\] done
clear
C)
\[\frac{\sqrt{e}+1}{e}\] done
clear
D)
\[\frac{\sqrt{e}-1}{e}\] done
clear
View Solution play_arrow
-
question_answer9)
\[\int_{0}^{1}{{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\,dx=}\] [Karnataka CET 1999]
A)
\[\frac{\pi }{2}-2\log \sqrt{2}\] done
clear
B)
\[\frac{\pi }{2}+2\log \sqrt{2}\] done
clear
C)
\[\frac{\pi }{4}-\log \sqrt{2}\] done
clear
D)
\[\frac{\pi }{4}+\log \sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer10)
\[\int_{1}^{e}{\frac{{{e}^{x}}}{x}(1+x\log x)\,dx}=\]
A)
\[{{e}^{e}}\] done
clear
B)
\[{{e}^{e}}-e\] done
clear
C)
\[{{e}^{e}}+e\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer11)
The value of \[\int_{-2}^{2}{(a{{x}^{3}}+bx+c)}\] depends on [MNR 1988; UPSEAT 2000]
A)
The value of \[a\] done
clear
B)
The value of \[b\] done
clear
C)
The value of \[c\] done
clear
D)
The values of \[a\] and \[b\] done
clear
View Solution play_arrow
-
question_answer12)
\[\int_{\pi /6}^{\pi /4}{\text{cosec}\,2x\,dx=}\] [MNR 1980]
A)
\[\log 3\] done
clear
B)
\[\log \sqrt{3}\] done
clear
C)
\[\log 9\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer13)
\[\int_{0}^{\pi /2}{\sqrt{\cos \theta }{{\sin }^{3}}\theta }\,d\theta =\]
A)
\[\frac{20}{21}\] done
clear
B)
\[\frac{8}{21}\] done
clear
C)
\[\frac{-20}{21}\] done
clear
D)
\[\frac{-8}{21}\] done
clear
View Solution play_arrow
-
question_answer14)
\[\int_{0}^{\pi /4}{{{\sec }^{7}}\theta {{\sin }^{3}}\theta }\,d\theta =\]
A)
\[\frac{1}{12}\] done
clear
B)
\[\frac{3}{12}\] done
clear
C)
\[\frac{5}{12}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer15)
\[\int_{a}^{b}{\frac{\log x}{x}\,dx=}\] [MP PET 1994]
A)
\[\log \left( \frac{\log b}{\log a} \right)\] done
clear
B)
\[\log (a\,b)\log \,\left( \frac{b}{a} \right)\] done
clear
C)
\[\frac{1}{2}\log (a\,b)\log \,\left( \frac{b}{a} \right)\] done
clear
D)
\[\frac{1}{2}\log (a\,b)\log \,\left( \frac{a}{b} \right)\] done
clear
View Solution play_arrow
-
question_answer16)
\[\int_{0}^{1}{{{\tan }^{-1}}x\,dx=}\] [Karnataka CET 1993; RPET 1997]
A)
\[\frac{\pi }{4}-\frac{1}{2}\log 2\] done
clear
B)
\[\pi -\frac{1}{2}\log 2\] done
clear
C)
\[\frac{\pi }{4}-\log 2\] done
clear
D)
\[\pi -\log 2\] done
clear
View Solution play_arrow
-
question_answer17)
\[\int_{0}^{1}{\frac{dx}{{{[ax+b(1-x)]}^{2}}}}=\] [SCRA 1986]
A)
\[\frac{a}{b}\] done
clear
B)
\[\frac{b}{a}\] done
clear
C)
\[a\,b\] done
clear
D)
\[\frac{1}{a\,b}\] done
clear
View Solution play_arrow
-
question_answer18)
If \[\int_{0}^{k}{\frac{dx}{2+8{{x}^{2}}}}=\frac{\pi }{16}\,,\] then \[k=\]
A)
1 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer19)
\[\int_{\pi /4}^{\pi /2}{\cos \theta \,\text{cose}{{\text{c}}^{\text{2}}}\theta \,d\theta =}\] [Roorkee 1978]
A)
\[\sqrt{2}-1\] done
clear
B)
\[1-\sqrt{2}\] done
clear
C)
\[\sqrt{2}+1\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer20)
\[\int_{0}^{1/\sqrt{2}}{\frac{{{\sin }^{-1}}x}{{{(1-{{x}^{2}})}^{3/2}}}dx=}\] [Roorkee 1984]
A)
\[\frac{\pi }{4}+\frac{1}{2}\log 2\] done
clear
B)
\[\frac{\pi }{4}-\frac{1}{2}\log 2\] done
clear
C)
\[\frac{\pi }{2}+\log 2\] done
clear
D)
\[\frac{\pi }{2}-\log 2\] done
clear
View Solution play_arrow
-
question_answer21)
The correct evaluation of \[\int_{0}^{\pi /2}{\sin x\,\sin 2x}\] is [MP PET 1993, 2003]
A)
\[\frac{4}{3}\] done
clear
B)
\[\frac{1}{3}\] done
clear
C)
\[\frac{3}{4}\] done
clear
D)
\[\frac{2}{3}\] done
clear
View Solution play_arrow
-
question_answer22)
\[\int_{0}^{\pi /2}{\frac{dx}{2+\cos x}}=\] [BIT Ranchi 1992]
A)
\[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\] done
clear
B)
\[\sqrt{3}{{\tan }^{-1}}\left( \sqrt{3} \right)\] done
clear
C)
\[\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\] done
clear
D)
\[2\sqrt{3}{{\tan }^{-1}}\left( \sqrt{3} \right)\] done
clear
View Solution play_arrow
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question_answer23)
\[\int_{0}^{1}{\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}}\,dx=\] [SCRA 1987; MNR 1990]
A)
\[\frac{{{\pi }^{2}}}{8}\] done
clear
B)
\[\frac{{{\pi }^{2}}}{16}\] done
clear
C)
\[\frac{{{\pi }^{2}}}{4}\] done
clear
D)
\[\frac{{{\pi }^{2}}}{32}\] done
clear
View Solution play_arrow
-
question_answer24)
\[\int_{1}^{2}{\frac{\cos (\log x)}{x}}\,dx=\]
A)
\[\sin \,(\log 3)\] done
clear
B)
\[\sin \,(\log 2)\] done
clear
C)
\[\cos \,(\log 3)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer25)
\[\int_{0}^{a}{\frac{x\,dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}=\]
A)
\[a\,(\sqrt{2}-1)\] done
clear
B)
\[a\,(1-\sqrt{2})\] done
clear
C)
\[a\,(1+\sqrt{2})\] done
clear
D)
\[2a\sqrt{3}\] done
clear
View Solution play_arrow
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question_answer26)
The value of integral \[\int_{1/\pi }^{2/\pi }{\frac{\sin (1/x)}{{{x}^{2}}}}\,dx=\] [IIT 1990]
A)
2 done
clear
B)
\[-1\] done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer27)
\[\int_{0}^{a}{\frac{{{x}^{4}}\,dx}{{{({{a}^{2}}+{{x}^{2}})}^{4}}}}=\]
A)
\[\frac{1}{16{{a}^{3}}}\left( \frac{\pi }{4}-\frac{1}{3} \right)\] done
clear
B)
\[\frac{1}{16{{a}^{3}}}\left( \frac{\pi }{4}+\frac{1}{3} \right)\] done
clear
C)
\[\frac{1}{16}{{a}^{3}}\left( \frac{\pi }{4}-\frac{1}{3} \right)\] done
clear
D)
\[\frac{1}{16}{{a}^{3}}\left( \frac{\pi }{4}+\frac{1}{3} \right)\] done
clear
View Solution play_arrow
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question_answer28)
\[\int_{0}^{2\pi }{{{e}^{x/2}}.\sin \left( \frac{x}{2}+\frac{\pi }{4} \right)\,dx=}\] [Roorkee 1982]
A)
1 done
clear
B)
\[2\sqrt{2}\] done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer29)
\[\int_{0}^{1}{\frac{{{e}^{-x}}}{1+{{e}^{-x}}}}\,dx=\] [Roorkee 1976]
A)
\[\log \left( \frac{1+e}{e} \right)-\frac{1}{e}+1\] done
clear
B)
\[\log \left( \frac{1+e}{2e} \right)-\frac{1}{e}+1\] done
clear
C)
\[\log \left( \frac{1+e}{2e} \right)+\frac{1}{e}-1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer30)
\[\int_{0}^{\pi /4}{\frac{\sin x+\cos x}{9+16\sin 2x}\,dx=}\] [IIT 1983]
A)
\[\frac{1}{20}\log 3\] done
clear
B)
\[\log 3\] done
clear
C)
\[\frac{1}{20}\log 5\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer31)
\[\int_{\pi \text{/4}}^{\pi \text{/2}}{{{e}^{x}}(\log \sin x+\cot x)\,dx=}\] [AI CBSE 1991]
A)
\[{{e}^{\pi /4}}\log 2\] done
clear
B)
\[-{{e}^{\pi /4}}\log 2\] done
clear
C)
\[\frac{1}{2}{{e}^{\pi /4}}\log 2\] done
clear
D)
\[-\frac{1}{2}{{e}^{\pi /4}}\log 2\] done
clear
View Solution play_arrow
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question_answer32)
\[\int_{0}^{1/2}{\frac{x{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\,dx=}\] [IIT 1984]
A)
\[\frac{1}{2}+\frac{\sqrt{3}\pi }{12}\] done
clear
B)
\[\frac{1}{2}-\frac{\sqrt{3}\pi }{12}\] done
clear
C)
\[\frac{1}{2}-\frac{\sqrt{3\pi }}{12}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer33)
\[\int_{0}^{2}{\sqrt{\frac{2+x}{2-x}}}\,dx=\] [MNR 1984; CEE 1993]
A)
\[\pi +2\] done
clear
B)
\[\pi +\frac{3}{2}\] done
clear
C)
\[\pi +1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
\[\int_{0}^{\pi }{\frac{dx}{1+\sin x}}=\] [CEE 1993]
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
2 done
clear
D)
\[\frac{3}{2}\] done
clear
View Solution play_arrow
-
question_answer35)
\[\int_{0}^{\pi /8}{\frac{{{\sec }^{2}}2x}{2}\,dx=}\]
A)
\[\frac{1}{4}\] done
clear
B)
\[\frac{1}{3}\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer36)
\[\int_{0}^{2\pi }{\sqrt{1+\sin \frac{x}{2}}\,dx=}\] [MNR 1987; UPSEAT 2000]
A)
0 done
clear
B)
2 done
clear
C)
8 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer37)
\[\int_{0}^{1}{{{\cos }^{-1}}x\,dx=}\] [DSSE 1988]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer38)
\[\int_{0}^{\pi /2}{\frac{\cos x}{1+\cos x+\sin x}}\,dx=\] [Roorkee 1989]
A)
\[\frac{\pi }{4}+\frac{1}{2}\log 2\] done
clear
B)
\[\frac{\pi }{4}+\log 2\] done
clear
C)
\[\frac{\pi }{4}-\frac{1}{2}\log 2\] done
clear
D)
\[\frac{\pi }{4}-\log 2\] done
clear
View Solution play_arrow
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question_answer39)
\[\int_{0}^{\pi /6}{(2+3{{x}^{2}})\cos 3x\,dx=}\] [DSSE 1985]
A)
\[\frac{1}{36}(\pi +16)\] done
clear
B)
\[\frac{1}{36}(\pi -16)\] done
clear
C)
\[\frac{1}{36}({{\pi }^{2}}-16)\] done
clear
D)
\[\frac{1}{36}({{\pi }^{2}}+16)\] done
clear
View Solution play_arrow
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question_answer40)
\[\int_{0}^{\pi /2}{\frac{\sin x\cos x}{1+{{\sin }^{4}}x}\,dx=}\] [AISSE 1988]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{6}\] done
clear
D)
\[\frac{\pi }{8}\] done
clear
View Solution play_arrow
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question_answer41)
\[\int_{0}^{\pi /4}{{{\tan }^{6}}x{{\sec }^{2}}x\,dx=}\] [BIT Ranchi 1981]
A)
\[\frac{1}{7}\] done
clear
B)
\[\frac{2}{7}\] done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer42)
\[\int_{0}^{2}{\frac{{{x}^{3}}\,dx}{{{({{x}^{2}}+1)}^{\frac{3}{2}}}}}=\]
A)
\[{{(\sqrt{2}-1)}^{2}}\] done
clear
B)
\[\frac{{{(\sqrt{2}-1)}^{2}}}{\sqrt{2}}\] done
clear
C)
\[\frac{\sqrt{2}-1}{\sqrt{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer43)
\[\int_{0}^{\pi /6}{\frac{\sin x}{{{\cos }^{3}}x}\,dx=}\] [SCRA 1979]
A)
\[\frac{2}{3}\] done
clear
B)
\[\frac{1}{6}\] done
clear
C)
2 done
clear
D)
\[\frac{1}{3}\] done
clear
View Solution play_arrow
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question_answer44)
\[\int_{0}^{\pi /2}{\frac{\sin x\cos x\,dx}{{{\cos }^{2}}x+3\cos x+2}}=\] [MNR 1981]
A)
\[\log \left( \frac{8}{9} \right)\] done
clear
B)
\[\log \left( \frac{9}{8} \right)\] done
clear
C)
\[\log (8\times 9)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer45)
The value of the integral \[\int_{0}^{\log 5}{\frac{{{e}^{x}}\sqrt{{{e}^{x}}-1}}{{{e}^{x}}+3}}\,dx=\]
A)
\[3+2\pi \] done
clear
B)
\[4-\pi \] done
clear
C)
\[2+\pi \] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer46)
The value of the definite integral \[\int_{0}^{1}{\frac{dx}{{{x}^{2}}+2x\cos \alpha +1}}\] for \[0<\alpha <\pi \] is equal to [Kurukshetra CEE 2002]
A)
\[\sin \alpha \] done
clear
B)
\[{{\tan }^{-1}}(\sin \alpha )\] done
clear
C)
\[\alpha \sin \alpha \] done
clear
D)
\[\frac{\alpha }{2}{{(\sin \alpha )}^{-1}}\] done
clear
View Solution play_arrow
-
question_answer47)
The value of the integral \[\int_{-\pi }^{\pi }{\sin mx\sin nx\,dx}\] for \[m\ne n\] \[(m,\,\,n\in I),\] is
A)
0 done
clear
B)
\[\pi \] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[2\pi \] done
clear
View Solution play_arrow
-
question_answer48)
The greater of \[\int_{0}^{\pi /2}{\frac{\sin x}{x}\,dx}\] and \[\frac{\pi }{2},\] is
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\int_{0}^{\pi /2}{\frac{\sin x}{x}\,dx}\] done
clear
C)
Nothing can be said done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer49)
The integral \[\int_{-1}^{3}{\left( {{\tan }^{-1}}\frac{x}{{{x}^{2}}+1}+{{\tan }^{-1}}\frac{{{x}^{2}}+1}{x} \right)}\,dx=\] [Karnataka CET 2000]
A)
\[\pi \] done
clear
B)
\[2\pi \] done
clear
C)
\[3\pi \] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer50)
If \[{{I}_{1}}=\int_{e}^{{{e}^{2}}}{\frac{dx}{\log x}}\] and \[{{I}_{2}}=\int_{1}^{2}{\frac{{{e}^{x}}}{x}\,dx,}\] then [Karnataka CET 2000]
A)
\[{{I}_{1}}={{I}_{2}}\] done
clear
B)
\[{{I}_{1}}>{{I}_{2}}\] done
clear
C)
\[{{I}_{1}}<{{I}_{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer51)
\[\int_{-\pi /4}^{\pi /2}{{{e}^{-x}}\sin x\,dx}=\] [CEE 1993]
A)
\[-\frac{1}{2}{{e}^{-\pi /2}}\] done
clear
B)
\[-\frac{\sqrt{2}}{2}{{e}^{-\pi /4}}\] done
clear
C)
\[-\sqrt{2}({{e}^{-\pi /4}}+{{e}^{-\pi /4}})\] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer52)
\[\int_{0}^{\pi /2}{\frac{1+2\cos x}{{{(2+\cos x)}^{2}}}=}\] [CEE 1993]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\pi \] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer53)
\[\int_{0}^{\pi }{\frac{dx}{1-2a\cos x+{{a}^{2}}}}\,\]= [CEE 1993]
A)
\[\frac{\pi }{2(1-{{a}^{2}})}\] done
clear
B)
\[\pi (1-{{a}^{2}})\] done
clear
C)
\[\frac{\pi }{1-{{a}^{2}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer54)
\[\int_{0}^{1}{{{(1-x)}^{9}}dx=}\] [SCRA 1979]
A)
1 done
clear
B)
\[\frac{1}{10}\] done
clear
C)
\[\frac{11}{10}\] done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer55)
\[\int_{0}^{\pi /3}{\cos 3x\,dx=}\] [SCRA 1980]
A)
\[\pi \] done
clear
B)
0 done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\frac{\pi }{4}\] done
clear
View Solution play_arrow
-
question_answer56)
The value of \[\int_{0}^{\pi /4}{\frac{1+\tan x}{1-\tan x}\,dx}\] is [SCRA 1986]
A)
\[-\frac{1}{2}\log 2\] done
clear
B)
\[\frac{1}{4}\log 2\] done
clear
C)
\[\frac{1}{3}\log 2\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer57)
The value of \[\int_{0}^{1}{\frac{dx}{{{e}^{x}}+{{e}^{-x}}}}\] is [SCRA 1980]
A)
\[{{\tan }^{-1}}\left( \frac{1-e}{1+e} \right)\] done
clear
B)
\[{{\tan }^{-1}}\left( \frac{e-1}{e+1} \right)\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[{{\tan }^{-1}}e+\frac{\pi }{4}\] done
clear
View Solution play_arrow
-
question_answer58)
\[\int_{1}^{e}{\frac{1+\log x}{x}\,dx=}\] [SCRA 1986]
A)
\[\frac{3}{2}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[\frac{1}{e}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer59)
If \[\int_{0}^{1}{x\log \left( 1+\frac{x}{2} \right)}\,dx=a+b\log \frac{2}{3},\] then [SCRA 1986]
A)
\[a=\frac{3}{2},\,\,\,b=\frac{3}{2}\] done
clear
B)
\[a=\frac{3}{4},\,\,\,b=-\frac{3}{4}\] done
clear
C)
\[a=\frac{3}{4},\,\,\,b=\frac{3}{2}\] done
clear
D)
\[a=b\] done
clear
View Solution play_arrow
-
question_answer60)
\[\int_{0}^{1}{\frac{dx}{\sqrt{1+x}-\sqrt{x}}=}\] [SCRA 1986]
A)
\[\frac{2\sqrt{2}}{3}\] done
clear
B)
\[\frac{4\sqrt{2}}{3}\] done
clear
C)
\[\frac{8\sqrt{2}}{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer61)
\[\int_{0}^{\pi /4}{\frac{4\sin 2\theta \,d\theta }{{{\sin }^{4}}\theta +{{\cos }^{4}}\theta }}=\] [SCRA 1986]
A)
\[\pi /4\] done
clear
B)
\[\pi /2\] done
clear
C)
\[\pi \] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer62)
\[\int_{0}^{1}{\frac{{{e}^{x}}(x-1)}{{{(x+1)}^{3}}}\,dx=}\] [SCRA 1986]
A)
\[\frac{e}{4}\] done
clear
B)
\[\frac{e}{4}-1\] done
clear
C)
\[\frac{e}{4}+1\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer63)
If \[x({{x}^{4}}+1)\varphi (x)=1,\] then \[\int_{1}^{2}{\varphi (x)\,dx=}\] [SCRA 1986]
A)
\[\frac{1}{4}\log \frac{32}{17}\] done
clear
B)
\[\frac{1}{2}\log \frac{32}{17}\] done
clear
C)
\[\frac{1}{4}\log \frac{16}{17}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer64)
\[\int_{1/4}^{1/2}{\frac{dx}{\sqrt{x-{{x}^{2}}}}=}\] [SCRA 1986]
A)
\[\pi \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{3}\] done
clear
D)
\[\frac{\pi }{6}\] done
clear
View Solution play_arrow
-
question_answer65)
The value of \[\int_{0}^{2}{\frac{{{3}^{\sqrt{x}}}}{\sqrt{x}}}\,dx\] is [SCRA 1992]
A)
\[\frac{2}{\log 3}.({{3}^{\sqrt{2}}}-1)\] done
clear
B)
0 done
clear
C)
\[2.\frac{\sqrt{2}}{\log 3}\] done
clear
D)
\[\frac{{{3}^{\sqrt{2}}}}{\sqrt{2}}\] done
clear
View Solution play_arrow
-
question_answer66)
\[\int_{0}^{2\pi }{\,\,(\sin x+\cos x)\,dx=}\] [SCRA 1991]
A)
0 done
clear
B)
2 done
clear
C)
\[-2\] done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer67)
\[\int_{0}^{\pi /4}{\frac{\sec x}{1+2{{\sin }^{2}}x}}\] is equal to [MNR 1994]
A)
\[\frac{1}{3}\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\] done
clear
B)
\[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\] done
clear
C)
\[3\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\] done
clear
D)
\[3\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\] done
clear
View Solution play_arrow
-
question_answer68)
The value of \[\int_{0}^{\pi /2}{\frac{\sin x}{1+{{\cos }^{2}}x}\,dx}\] is [RPET 1995]
A)
\[\pi /2\] done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /3\] done
clear
D)
\[\pi /6\] done
clear
View Solution play_arrow
-
question_answer69)
The value of \[\int_{1}^{2}{\log x\,dx}\] is [Roorkee 1995]
A)
\[\log 2/e\] done
clear
B)
\[\log 4\] done
clear
C)
\[\log 4/e\] done
clear
D)
\[\log 2\] done
clear
View Solution play_arrow
-
question_answer70)
The value of \[\int_{3}^{5}{\frac{{{x}^{2}}}{{{x}^{2}}-4}\,dx}\] is [Roorkee 1992]
A)
\[2-{{\log }_{e}}\left( \frac{15}{7} \right)\] done
clear
B)
\[2+{{\log }_{e}}\left( \frac{15}{7} \right)\] done
clear
C)
\[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\] done
clear
D)
\[2-{{\tan }^{-1}}\left( \frac{15}{7} \right)\] done
clear
View Solution play_arrow
-
question_answer71)
\[\int_{0}^{\pi /4}{\frac{dx}{{{\cos }^{4}}x-{{\cos }^{2}}x{{\sin }^{2}}x+{{\sin }^{4}}x}=}\]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer72)
The value of \[\int_{0}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t}\,dt+\int_{0}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}\,dt}}\] is [MP PET 2001; Orissa JEE 2005]
A)
\[\frac{\pi }{2}\] done
clear
B)
1 done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer73)
If for non-zero \[x,\] \[af(x)+bf\left( \frac{1}{x} \right)=\frac{1}{x}-5,\] where \[a\ne b,\] then \[\int_{1}^{2}{f(x)\,dx=}\] [IIT 1996]
A)
\[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\] done
clear
B)
\[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a+\frac{7}{2}b \right]\] done
clear
C)
\[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\] done
clear
D)
\[\frac{1}{({{a}^{2}}+{{b}^{2}})}\left[ a\log 2-5a-\frac{7}{2}b \right]\] done
clear
View Solution play_arrow
-
question_answer74)
If \[{{I}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta \,d\theta ,}\] then \[{{I}_{8}}+{{I}_{6}}\] equals [Kurukshetra CEE 1996]
A)
\[\frac{1}{4}\] done
clear
B)
\[\frac{1}{5}\] done
clear
C)
\[\frac{1}{6}\] done
clear
D)
\[\frac{1}{7}\] done
clear
View Solution play_arrow
-
question_answer75)
\[\int_{0}^{\pi /4}{{}}\sec x\log (\sec x+\tan x)\,dx=\]
A)
\[\frac{1}{2}{{[\log (1+\sqrt{2})]}^{2}}\] done
clear
B)
\[{{[\log (1+\sqrt{2})]}^{2}}\] done
clear
C)
\[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\] done
clear
D)
\[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\] done
clear
View Solution play_arrow
-
question_answer76)
\[\int_{0}^{\pi /4}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}}\,dx=\]
A)
\[{{\log }_{e}}\left( \frac{2}{3} \right)\] done
clear
B)
\[{{\log }_{e}}3\] done
clear
C)
\[\frac{1}{2}{{\log }_{e}}\left( \frac{4}{3} \right)\] done
clear
D)
\[{{\log }_{e}}\left( \frac{4}{3} \right)\] done
clear
View Solution play_arrow
-
question_answer77)
\[\int_{0}^{2/3}{\frac{dx}{4+9{{x}^{2}}}=}\] [MP PET 1997]
A)
\[\frac{\pi }{12}\] done
clear
B)
\[\frac{\pi }{24}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer78)
The value of \[\int_{0}^{1}{\frac{{{x}^{4}}+1}{{{x}^{2}}+1}\,dx}\] is [MP PET 1998]
A)
\[\frac{1}{6}(3\pi -4)\] done
clear
B)
\[\frac{1}{6}(3-4\pi )\] done
clear
C)
\[\frac{1}{6}(3\pi +4)\] done
clear
D)
\[\frac{1}{6}(3+4\pi )\] done
clear
View Solution play_arrow
-
question_answer79)
\[\int_{0}^{a}{{{x}^{2}}\sin {{x}^{3}}\,dx}\] equals [RPET 1996]
A)
\[(1-\cos {{a}^{3}})\] done
clear
B)
\[3(1-\cos {{a}^{3}})\] done
clear
C)
\[-\frac{1}{3}(1-\cos {{a}^{3}})\] done
clear
D)
\[\frac{1}{3}(1-\cos {{a}^{3}})\] done
clear
View Solution play_arrow
-
question_answer80)
\[\int_{0}^{\pi /4}{[\sqrt{\tan x}+\sqrt{\cot x}]\,dx}\] equals [RPET 1997]
A)
\[\sqrt{2}\pi \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{\sqrt{2}}\] done
clear
D)
\[2\pi \] done
clear
View Solution play_arrow
-
question_answer81)
\[\int_{0}^{1}{\sqrt{\frac{1-x}{1+x}}}\,dx\] equals [RPET 1997; IIT Screening 2004]
A)
\[\left( \frac{\pi }{2}-1 \right)\] done
clear
B)
\[\left( \frac{\pi }{2}+1 \right)\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[(\pi +1)\] done
clear
View Solution play_arrow
-
question_answer82)
\[\int_{1}^{e}{\frac{1}{x}\,dx}\] is equals to [SCRA 1996; Pb. CET 2003]
A)
\[\infty \] done
clear
B)
0 done
clear
C)
1 done
clear
D)
\[\log (1+e)\] done
clear
View Solution play_arrow
-
question_answer83)
\[\int_{\,1}^{\,x}{\frac{\log {{x}^{2}}}{x}\,dx=}\] [DCE 1999]
A)
\[{{(\log x)}^{2}}\] done
clear
B)
\[\frac{1}{2}{{(\log x)}^{2}}\] done
clear
C)
\[\frac{\log {{x}^{2}}}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer84)
\[\frac{1}{2}(e-3)\] [SCRA 1986; Karnataka CET 1999]
A)
\[\pi ab\] done
clear
B)
\[{{\pi }^{2}}ab\] done
clear
C)
\[\frac{\pi }{ab}\] done
clear
D)
\[\frac{\pi }{2ab}\] done
clear
View Solution play_arrow
-
question_answer85)
\[\int_{0}^{\pi /4}{{}}(\cos x-\sin x)dx+\int_{\pi /4}^{5\pi /4}{{}}(\sin x-\cos x)dx\] \[+\int_{2\pi }^{\pi /4}{{}}(\cos x-\sin x)\,dx\] is equal to [RPET 2000]
A)
\[\sqrt{2}-2\] done
clear
B)
\[2\sqrt{2}-2\] done
clear
C)
\[3\sqrt{2}-2\] done
clear
D)
\[4\sqrt{2}-2\] done
clear
View Solution play_arrow
-
question_answer86)
\[\left( \int_{\,0}^{\,a}{x\,dx} \right)\le (a+4),\] then [RPET 2000]
A)
\[\frac{24}{5}\pi \] done
clear
B)
\[-2\le a\le 4\] done
clear
C)
\[-2\le a\le 0\] done
clear
D)
\[a\le -2\,\,\text{or}\,\,a\ge 4\] done
clear
View Solution play_arrow
-
question_answer87)
The value of \[\int_{\,-\,1}^{\,3}{\,{{\tan }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right)+{{\tan }^{-1}}\left( \frac{{{x}^{2}}+1}{x} \right)\,dx}\] is [Karnataka CET 2000]
A)
\[2\pi \] done
clear
B)
\[\pi \] done
clear
C)
\[\frac{21}{5}\pi \] done
clear
D)
\[\frac{\pi }{4}\] done
clear
View Solution play_arrow
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question_answer88)
\[\int_{\,-\,1}^{\,0}{\frac{dx}{{{x}^{2}}+2x+2}=}\] [MP PET 2000]
A)
0 done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /2\] done
clear
D)
\[-\pi /4\] done
clear
View Solution play_arrow
-
question_answer89)
The value of \[\int_{\,0}^{\,1}{\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}dx}\] is [RPET 2001]
A)
\[\pi /4\] done
clear
B)
\[{{\pi }^{2}}/32\] done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer90)
\[\int_{1}^{\sqrt{3}}{\frac{1}{1+{{x}^{2}}}dx}\] is equal to [DCE 2002]
A)
\[\pi /12\] done
clear
B)
\[\pi /6\] done
clear
C)
\[\pi /4\] done
clear
D)
\[\pi /3\] done
clear
View Solution play_arrow
-
question_answer91)
\[\int_{\,1}^{\,3}{(x-1)(x-2)(x-3)dx=}\] [Karnataka CET 2002]
A)
3 done
clear
B)
2 done
clear
C)
1 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer92)
\[\int_{\,2}^{\,3}{\frac{dx}{{{x}^{2}}-x}=}\] [EAMCET 2002]
A)
\[\log (2/3)\] done
clear
B)
\[\log (1/4)\] done
clear
C)
\[\log (4/3)\] done
clear
D)
\[\log (8/3)\] done
clear
View Solution play_arrow
-
question_answer93)
\[\int_{\,8}^{\,15}{\frac{dx}{(x-3)\sqrt{x+1}}=}\] [UPSEAT 2003]
A)
\[\frac{1}{2}\log \frac{5}{3}\] done
clear
B)
\[\frac{1}{3}\log \frac{5}{3}\] done
clear
C)
\[\frac{1}{2}\log \frac{3}{5}\] done
clear
D)
\[\frac{1}{5}\log \frac{3}{5}\] done
clear
View Solution play_arrow
-
question_answer94)
The value of \[\int_{\,0}^{\,\pi }{\,\left| \,{{\sin }^{3}}\theta \, \right|\,d\theta }\] is [UPSEAT 2003]
A)
0 done
clear
B)
3/8 done
clear
C)
4/3 done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
-
question_answer95)
\[\int_{\,0}^{\,1}{\,\sin \left( 2{{\tan }^{-1}}\sqrt{\frac{1+x}{1-x}} \right)\,dx=}\] [EAMCET 2003]
A)
\[\pi /6\] done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /2\] done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
-
question_answer96)
\[\int_{\,0}^{\,3}{\,\frac{3x+1}{{{x}^{2}}+9}dx=}\] [EAMCET 2003]
A)
\[\log (2\sqrt{2})+\frac{\pi }{12}\] done
clear
B)
\[\log (2\sqrt{2})+\frac{\pi }{2}\] done
clear
C)
\[\log (2\sqrt{2})+\frac{\pi }{6}\] done
clear
D)
\[\log (2\sqrt{2})+\frac{\pi }{3}\] done
clear
View Solution play_arrow
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question_answer97)
The value of \[\int_{1}^{2}{\frac{dx}{x(1+{{x}^{4}})}}\]is [MP PET 2004]
A)
\[\frac{1}{4}\log \frac{17}{32}\] done
clear
B)
\[\frac{1}{4}\log \frac{17}{2}\] done
clear
C)
\[\log \frac{17}{2}\] done
clear
D)
\[\frac{1}{4}\log \frac{32}{17}\] done
clear
View Solution play_arrow
-
question_answer98)
The value of\[\int_{2}^{3}{\frac{x+1}{{{x}^{2}}(x-1)}dx}\]is [MP PET 2004]
A)
\[2\log 2-\frac{1}{6}\] done
clear
B)
\[\log \frac{16}{9}-\frac{1}{6}\] done
clear
C)
\[\log \frac{4}{3}-\frac{1}{6}\] done
clear
D)
\[\log \frac{16}{9}+\frac{1}{6}\] done
clear
View Solution play_arrow
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question_answer99)
The value of \[\int_{1}^{e}{\log x\,dx}\]is [Pb. CET 2001]
A)
\[0\] done
clear
B)
1 done
clear
C)
\[e-1\] done
clear
D)
\[e+1\] done
clear
View Solution play_arrow
-
question_answer100)
The value of \[I=\int_{0}^{\pi /2}{\frac{{{(\sin x+\cos x)}^{2}}}{\sqrt{1+\sin 2x}}\text{ }}dx\]is [AIEEE 2004]
A)
3 done
clear
B)
1 done
clear
C)
2 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer101)
\[\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta d\theta }=\] [Karnataka CET 2004]
A)
\[\frac{2}{3}\] done
clear
B)
\[\frac{1}{4}\] done
clear
C)
\[\frac{1}{3}\] done
clear
D)
\[\frac{1}{6}\] done
clear
View Solution play_arrow
-
question_answer102)
\[\int_{3}^{8}{\frac{2-3x}{x\sqrt{(1+x)}}\text{ }}dx\]is equal to [Pb. CET 2001]
A)
\[2\log \,\left( 3/2{{e}^{3}} \right)\] done
clear
B)
\[\log (3/{{e}^{3}})\] done
clear
C)
\[4\log (3/{{e}^{3}})\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer103)
The value of \[\int_{0}^{1}{{{x}^{2}}{{e}^{x}}dx}\]is equal to [Pb. CET 2002]
A)
\[e-2\] done
clear
B)
\[e+2\] done
clear
C)
\[{{e}^{2}}-2\] done
clear
D)
\[{{e}^{2}}\] done
clear
View Solution play_arrow
-
question_answer104)
Let \[{{I}_{1}}=\int_{1}^{2}{\frac{dx}{\sqrt{1+{{x}^{2}}}}}\]and\[{{I}_{2}}=\int_{1}^{2}{\frac{dx}{x}}\] then [Pb. CET 2004]
A)
\[{{I}_{1}}>{{I}_{2}}\] done
clear
B)
\[{{I}_{2}}>{{I}_{1}}\] done
clear
C)
\[{{I}_{1}}={{I}_{2}}\] done
clear
D)
\[{{I}_{1}}>2{{I}_{2}}\] done
clear
View Solution play_arrow
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question_answer105)
The value of \[\int_{1/e}^{\tan x}{\frac{t\,dt}{1+{{t}^{2}}}}+\int_{1/e}^{\cot x}{\frac{dt}{t(1+{{t}^{2}})}}=\] [IIT Screening]
A)
\[-1\] done
clear
B)
1 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer106)
\[\int\limits_{\pi /4}^{3\pi /4}{\frac{dx}{1+\cos x}}\] is equal to [IIT 1999]
A)
2 done
clear
B)
\[-2\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
\[-\frac{1}{2}\] done
clear
View Solution play_arrow
-
question_answer107)
The value of \[\int_{1}^{{{e}^{2}}}{\frac{dx}{x{{(1+\ln x)}^{2}}}}\] is [J & K 2005]
A)
\[2/3\] done
clear
B)
\[1/3\] done
clear
C)
\[3/2\] done
clear
D)
\[\ln 2\] done
clear
View Solution play_arrow
-
question_answer108)
\[\int_{\pi /4}^{\pi /2}{\text{cose}{{\text{c}}^{2}}xdx=}\] [Karnataka CET 2005]
A)
\[-1\] done
clear
B)
1 done
clear
C)
0 done
clear
D)
\[\frac{1}{2}\] done
clear
View Solution play_arrow
-
question_answer109)
If \[\int_{\log 2}^{x}{\frac{du}{{{({{e}^{u}}-1)}^{1/2}}}}=\frac{\pi }{6}\], then \[{{e}^{x}}=\] [Orissa JEE 2005]
A)
1 done
clear
B)
2 done
clear
C)
4 done
clear
D)
-1 done
clear
View Solution play_arrow
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question_answer110)
If \[g(1)=g(2)\], then \[\int_{1}^{2}{{{\left[ fg(x) \right]}^{-1}}}f'\{g(x)\}\ g'(x)\ dx\]is equal to [AMU 2005]
A)
1 done
clear
B)
2 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow