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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 3951. |
\[\int_{{}}^{{}}{\frac{x-1}{(x-3)(x-2)}dx=}\] [Roorkee 1978] |
| A. | \[\log (x-3)-\log (x-2)+c\] |
| B. | \[\log {{(x-3)}^{2}}-\log (x-2)+c\] |
| C. | \[\log (x-3)+\log (x-2)+c\] |
| D. | \[\log {{(x-3)}^{2}}+\log (x-2)+c\] |
| Answer» C. \[\log (x-3)+\log (x-2)+c\] | |
| 3952. |
\[\int{\frac{dx}{{{x}^{2}}+4x+13}}\] is equal to [Kerala (Engg.) 2002] |
| A. | \[\log ({{x}^{2}}+4x+13)+c\] |
| B. | \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{x+2}{3} \right)+c\] |
| C. | \[\log (2x+4)+c\] |
| D. | \[\frac{2x+4}{{{({{x}^{2}}+4x+13)}^{2}}}+c\] |
| Answer» C. \[\log (2x+4)+c\] | |
| 3953. |
\[\int{\frac{dx}{7+5\cos x}=}\] [EAMCET 2002] |
| A. | \[\frac{1}{\sqrt{6}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{6}}\tan \frac{x}{2} \right)+c\] |
| B. | \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] |
| C. | \[\frac{1}{4}{{\tan }^{-1}}\left( \tan \frac{x}{2} \right)+c\] |
| D. | \[\frac{1}{7}{{\tan }^{-1}}\left( \tan \frac{x}{2} \right)+c\] |
| Answer» B. \[\frac{1}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{3}}\tan \frac{x}{2} \right)+c\] | |
| 3954. |
\[\int_{{}}^{{}}{\frac{dx}{1+3{{\sin }^{2}}x}=}\] [Roorkee 1989; DCE 2001] |
| A. | \[\frac{1}{3}{{\tan }^{-1}}(3{{\tan }^{2}}x)+c\] |
| B. | \[\frac{1}{2}{{\tan }^{-1}}(2\tan x)+c\] |
| C. | \[{{\tan }^{-1}}(\tan x)+c\] |
| D. | None of these |
| Answer» C. \[{{\tan }^{-1}}(\tan x)+c\] | |
| 3955. |
\[\int_{{}}^{{}}{\frac{dx}{2{{x}^{2}}+x+1}}\ \]equals [RPET 1997] |
| A. | \[\frac{1}{\sqrt{7}}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] |
| B. | \[\frac{1}{2\sqrt{7}}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] |
| C. | \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{4x+1}{\sqrt{7}} \right)+c\] |
| D. | None of these |
| Answer» E. | |
| 3956. |
\[\int_{{}}^{{}}{\frac{1}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}dx=}\] |
| A. | \[\frac{1}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right) \right]+c\] |
| B. | \[\frac{1}{({{b}^{2}}-{{a}^{2}})}\left[ \frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right) \right]+c\] |
| C. | \[\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)+c\] |
| D. | \[\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right)-\frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)+c\] |
| Answer» B. \[\frac{1}{({{b}^{2}}-{{a}^{2}})}\left[ \frac{1}{b}{{\tan }^{-1}}\left( \frac{x}{b} \right)-\frac{1}{a}{{\tan }^{-1}}\left( \frac{x}{a} \right) \right]+c\] | |
| 3957. |
\[\int_{{}}^{{}}{\frac{1}{1+{{\cos }^{2}}x}dx}=\] |
| A. | \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\tan x)+c\] |
| B. | \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{1}{2}\tan x \right)+c\] |
| C. | \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}\left( \frac{1}{\sqrt{2}}\tan x \right)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3958. |
\[\int_{{}}^{{}}{\frac{dx}{5+4\cos x}=}\] [Roorkee 1983; RPET 1997] |
| A. | \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan x \right)+c\] |
| B. | \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan x \right)+c\] |
| C. | \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan \frac{x}{2} \right)+c\] |
| D. | \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan \frac{x}{2} \right)+c\] |
| Answer» D. \[\frac{1}{3}{{\tan }^{-1}}\left( \frac{1}{3}\tan \frac{x}{2} \right)+c\] | |
| 3959. |
\[\int_{{}}^{{}}{\frac{dx}{\sqrt{2x-{{x}^{2}}}}=}\] [MP PET 1991; Karnataka CET 2002] |
| A. | \[{{\cos }^{-1}}(x-1)+c\] |
| B. | \[{{\sin }^{-1}}(x-1)+c\] |
| C. | \[{{\cos }^{-1}}(1+x)+c\] |
| D. | \[{{\sin }^{-1}}(1-x)+c\] |
| Answer» C. \[{{\cos }^{-1}}(1+x)+c\] | |
| 3960. |
\[\int_{{}}^{{}}{\frac{x\ dx}{({{x}^{2}}-{{a}^{2}})({{x}^{2}}-{{b}^{2}})}=}\] [Roorkee 1976] |
| A. | \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\] |
| B. | \[\frac{1}{{{a}^{2}}-{{b}^{2}}}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\] |
| C. | \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{a}^{2}}}{{{x}^{2}}-{{b}^{2}}} \right)+c\] |
| D. | \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\] |
| Answer» D. \[\frac{1}{2({{a}^{2}}-{{b}^{2}})}\log \left( \frac{{{x}^{2}}-{{b}^{2}}}{{{x}^{2}}-{{a}^{2}}} \right)+c\] | |
| 3961. |
\[\int_{{}}^{{}}{\frac{dx}{1+x+{{x}^{2}}+{{x}^{3}}}=}\] [MP PET 1991] |
| A. | \[\log \sqrt{1+x}-\frac{1}{2}\log \sqrt{1+{{x}^{2}}}+\frac{1}{2}{{\tan }^{-1}}x+c\] |
| B. | \[\log \sqrt{1+x}-\log \sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\] |
| C. | \[\log \sqrt{1+{{x}^{2}}}-\log \sqrt{1+x}+\frac{1}{2}{{\tan }^{-1}}x+c\] |
| D. | \[\log \sqrt{1+x}+{{\tan }^{-1}}x+\log \sqrt{1+{{x}^{2}}}+c\] |
| Answer» B. \[\log \sqrt{1+x}-\log \sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\] | |
| 3962. |
\[\int_{{}}^{{}}{{{\sin }^{5}}x{{\cos }^{4}}x\ dx=}\] |
| A. | \[-\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\] |
| B. | \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\] |
| C. | \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x+\frac{1}{9}{{\cos }^{9}}x+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\] | |
| 3963. |
\[\int_{{}}^{{}}{\frac{x}{{{x}^{4}}-1}dx=}\] |
| A. | \[\frac{1}{4}\log \left[ \frac{{{x}^{2}}-1}{{{x}^{2}}+1} \right]+c\] |
| B. | \[\frac{1}{4}\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right]+c\] |
| C. | \[\frac{1}{2}\log \left[ \frac{{{x}^{2}}-1}{{{x}^{2}}+1} \right]+c\] |
| D. | \[\frac{1}{2}\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right]+c\] |
| Answer» B. \[\frac{1}{4}\log \left[ \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right]+c\] | |
| 3964. |
\[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}+1-2{{e}^{-x}}}=}\] |
| A. | \[\log ({{e}^{x}}-1)-\log ({{e}^{x}}+2)+c\] |
| B. | \[\frac{1}{2}\log ({{e}^{x}}-1)-\frac{1}{3}\log ({{e}^{x}}+2)+c\] |
| C. | \[\frac{1}{3}\log ({{e}^{x}}-1)-\frac{1}{3}\log ({{e}^{x}}+2)+c\] |
| D. | \[\frac{1}{3}\log ({{e}^{x}}-1)+\frac{1}{3}\log ({{e}^{x}}+2)+c\] |
| Answer» D. \[\frac{1}{3}\log ({{e}^{x}}-1)+\frac{1}{3}\log ({{e}^{x}}+2)+c\] | |
| 3965. |
\[\int_{{}}^{{}}{\frac{\cos x}{(1+\sin x)(2+\sin x)}\ dx=}\] [Roorkee 1979] |
| A. | \[\log [(1+\sin x)(2+\sin x)]+c\] |
| B. | \[\log \frac{2+\sin x}{1+\sin x}+c\] |
| C. | \[\log \frac{1+\sin x}{2+\sin x}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3966. |
\[\int_{{}}^{{}}{\sin 2x\cos 3x\ dx=}\] [Roorkee 1976] |
| A. | \[\frac{1}{2}\left( \cos x+\frac{1}{5}\cos 5x \right)+c\] |
| B. | \[\frac{1}{2}\left( \cos x-\frac{1}{5}\cos 5x \right)+c\] |
| C. | \[\cos x+\frac{1}{5}\cos 5x+c\] |
| D. | \[\cos x-\frac{1}{5}\cos 5x+c\] |
| Answer» C. \[\cos x+\frac{1}{5}\cos 5x+c\] | |
| 3967. |
\[\int_{{}}^{{}}{\frac{{{e}^{x}}}{(1+{{e}^{x}})(2+{{e}^{x}})}dx=}\] |
| A. | \[\log [(1+{{e}^{x}})(2+{{e}^{x}})]+c\] |
| B. | \[\log \left[ \frac{1+{{e}^{x}}}{2+{{e}^{x}}} \right]+c\] |
| C. | \[\log [(1+{{e}^{x}})\sqrt{2+{{e}^{x}}}]+c\] |
| D. | None of these |
| Answer» C. \[\log [(1+{{e}^{x}})\sqrt{2+{{e}^{x}}}]+c\] | |
| 3968. |
If \[\int_{{}}^{{}}{\sin 5x\cos 3x\ dx=-\frac{\cos 8x}{16}}+A\], then \[A=\] [MP PET 1992] |
| A. | \[\frac{\sin 2x}{16}+\]constant |
| B. | \[-\frac{\cos 2x}{4}+\]constant |
| C. | Constant |
| D. | None of these |
| Answer» C. Constant | |
| 3969. |
\[\int_{{}}^{{}}{{{\sin }^{3}}x{{\cos }^{2}}x\ dx=}\] |
| A. | \[\frac{{{\cos }^{5}}x}{5}-\frac{{{\cos }^{3}}x}{3}+c\] |
| B. | \[\frac{{{\cos }^{5}}x}{5}+\frac{{{\cos }^{3}}x}{3}+c\] |
| C. | \[\frac{{{\sin }^{5}}x}{5}-\frac{{{\sin }^{3}}x}{3}+c\] |
| D. | \[\frac{{{\sin }^{5}}x}{5}+\frac{{{\sin }^{3}}x}{3}+c\] |
| Answer» B. \[\frac{{{\cos }^{5}}x}{5}+\frac{{{\cos }^{3}}x}{3}+c\] | |
| 3970. |
\[\int_{{}}^{{}}{\frac{dx}{(x-{{x}^{2}})}=}\] [Roorkee 1982] |
| A. | \[\log x-\log (1-x)+c\] |
| B. | \[\log (1-{{x}^{2}})+c\] |
| C. | \[-\log x+\log (1-x)+c\] |
| D. | \[\log (x-{{x}^{2}})+c\] |
| Answer» B. \[\log (1-{{x}^{2}})+c\] | |
| 3971. |
\[\int_{{}}^{{}}{\frac{dx}{({{x}^{2}}+1)({{x}^{2}}+4)}=}\] [MP PET 1995] |
| A. | \[\frac{1}{3}{{\tan }^{-1}}x-\frac{1}{3}{{\tan }^{-1}}\frac{x}{2}+c\] |
| B. | \[\frac{1}{3}{{\tan }^{-1}}x+\frac{1}{3}{{\tan }^{-1}}\frac{x}{2}+c\] |
| C. | \[\frac{1}{3}{{\tan }^{-1}}x-\frac{1}{6}{{\tan }^{-1}}\frac{x}{2}+c\] |
| D. | \[{{\tan }^{-1}}x-2{{\tan }^{-1}}\frac{x}{2}+c\] |
| Answer» D. \[{{\tan }^{-1}}x-2{{\tan }^{-1}}\frac{x}{2}+c\] | |
| 3972. |
\[\int_{{}}^{{}}{\frac{1}{x-{{x}^{3}}}\ dx=}\] [MP PET 1996] |
| A. | \[\frac{1}{2}\log \frac{(1-{{x}^{2}})}{{{x}^{2}}}+c\] |
| B. | \[\log \frac{(1-x)}{x(1+x)}+c\] |
| C. | \[\log x(1-{{x}^{2}})+c\] |
| D. | \[\frac{1}{2}\log \frac{{{x}^{2}}}{(1-{{x}^{2}})}+c\] |
| Answer» E. | |
| 3973. |
The value of \[\int_{{}}^{{}}{\frac{\sin x}{{{\cos }^{2}}x}\ dx}\] is |
| A. | \[\sin x+k\] |
| B. | \[\tan x+k\] |
| C. | \[\sec x+k\] |
| D. | \[\tan x+\sec x+k\] |
| Answer» D. \[\tan x+\sec x+k\] | |
| 3974. |
\[\int_{{}}^{{}}{\frac{1}{1+{{\sin }^{2}}x}\ dx=}\] |
| A. | \[\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] |
| B. | \[\sqrt{2}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] |
| C. | \[-\frac{1}{\sqrt{2}}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] |
| D. | \[-\sqrt{2}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] |
| Answer» B. \[\sqrt{2}{{\tan }^{-1}}(\sqrt{2}\tan x)+k\] | |
| 3975. |
\[\int_{{}}^{{}}{\frac{{{x}^{5}}\ dx}{\sqrt{(1+{{x}^{3}})}}=}\] [IIT 1975] |
| A. | \[\frac{2}{3}\sqrt{(1+{{x}^{3}})}({{x}^{3}}+2)\] |
| B. | \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}-4)\] |
| C. | \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}+4)\] |
| D. | \[\frac{2}{9}\sqrt{(1+{{x}^{3}})}({{x}^{3}}-2)\] |
| Answer» E. | |
| 3976. |
\[\int_{{}}^{{}}{\frac{{{({{x}^{4}}-x)}^{1/4}}}{{{x}^{5}}}\ dx}\] is equal to |
| A. | \[\frac{4}{15}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] |
| B. | \[\frac{4}{5}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] |
| C. | \[\frac{4}{15}{{\left( 1+\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{4}{5}{{\left( 1-\frac{1}{{{x}^{3}}} \right)}^{5/4}}+c\] | |
| 3977. |
\[\int_{{}}^{{}}{{{e}^{x}}\sin ({{e}^{x}})}\ dx=\] [MP PET 1995] |
| A. | \[-\cos {{e}^{x}}+c\] |
| B. | \[\cos {{e}^{x}}+c\] |
| C. | \[-\text{cosec}\,{{e}^{x}}+c\] |
| D. | None of these |
| Answer» B. \[\cos {{e}^{x}}+c\] | |
| 3978. |
\[\int_{{}}^{{}}{\frac{\log (x+\sqrt{1+{{x}^{2}}})}{\sqrt{1+{{x}^{2}}}}\ dx=}\] |
| A. | \[\frac{1}{2}{{[\log (x+\sqrt{1+{{x}^{2}}})]}^{2}}+c\] |
| B. | \[\log {{(x+\sqrt{1+{{x}^{2}}})}^{2}}+c\] |
| C. | \[\log (x+\sqrt{1+{{x}^{2}}})+c\] |
| D. | None of these |
| Answer» B. \[\log {{(x+\sqrt{1+{{x}^{2}}})}^{2}}+c\] | |
| 3979. |
\[\int_{{}}^{{}}{\frac{1}{x{{\cos }^{2}}(1+\log x)}\ dx=}\] |
| A. | \[\tan \,(1+\log x)+c\] |
| B. | \[\cot \,(1+\log x)+c\] |
| C. | \[-\tan \,(1+\log x)+c\] |
| D. | \[-\cot (\,1+\log x)+c\] |
| Answer» B. \[\cot \,(1+\log x)+c\] | |
| 3980. |
\[\int_{{}}^{{}}{\frac{\sin 2x}{1+{{\sin }^{2}}x}dx=}\] [Roorkee 1976] |
| A. | \[\log \sin 2x+c\] |
| B. | \[\log (1+{{\sin }^{2}}x)+c\] |
| C. | \[\frac{1}{2}\log (1+{{\sin }^{2}}x)+c\] |
| D. | \[{{\tan }^{-1}}(\sin x)+c\] |
| Answer» C. \[\frac{1}{2}\log (1+{{\sin }^{2}}x)+c\] | |
| 3981. |
\[\int_{{}}^{{}}{\frac{1}{({{x}^{2}}-1)\sqrt{{{x}^{2}}+1}}}\ dx=\] |
| A. | \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}+x\sqrt{2}}{\sqrt{1+{{x}^{2}}}-x\sqrt{2}} \right\}+c\] |
| B. | \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}-\sqrt{2}}{\sqrt{1+{{x}^{2}}}+\sqrt{2}} \right\}+c\] |
| C. | \[\frac{1}{2\sqrt{2}}\log \left\{ \frac{\sqrt{1+{{x}^{2}}}-x\sqrt{2}}{\sqrt{1+{{x}^{2}}}+x\sqrt{2}} \right\}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3982. |
\[\int_{{}}^{{}}{\sqrt{\frac{x}{{{a}^{3}}-{{x}^{3}}}}\ dx=}\] |
| A. | \[{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] |
| B. | \[\frac{2}{3}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] |
| C. | \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] |
| D. | \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{2/3}}+c\] |
| Answer» C. \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] | |
| 3983. |
\[\int_{{}}^{{}}{\frac{\cos x-\sin x}{1+\sin 2x}\ dx=}\] [AISSE 1985] |
| A. | \[-\frac{1}{\cos x+\sin x}+c\] |
| B. | \[\frac{1}{\cos x+\sin x}+c\] |
| C. | \[\frac{1}{\cos x-\sin x}+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{\cos x+\sin x}+c\] | |
| 3984. |
\[\int_{{}}^{{}}{\frac{1+{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}dx=}\] [IIT 1977] |
| A. | \[\frac{3}{2}{{\sin }^{-1}}x-\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] |
| B. | \[\frac{3}{2}{{\sin }^{-1}}x+\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] |
| C. | \[\frac{3}{2}{{\cos }^{-1}}x-\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] |
| D. | \[\frac{3}{2}{{\cos }^{-1}}x+\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] |
| Answer» B. \[\frac{3}{2}{{\sin }^{-1}}x+\frac{1}{2}x\sqrt{1-{{x}^{2}}}+c\] | |
| 3985. |
\[\int_{{}}^{{}}{\frac{1+\tan x}{x+\log \sec x}\ dx=}\] [AI CBSE 1986] |
| A. | \[\log (x+\log \sec x)+c\] |
| B. | \[-\log (x+\log \sec x)+c\] |
| C. | \[\log (x-\log \sec x)+c\] |
| D. | None of these |
| Answer» B. \[-\log (x+\log \sec x)+c\] | |
| 3986. |
\[\int_{{}}^{{}}{\frac{(x+1){{(x+\log x)}^{2}}}{x}dx=}\] [AI CBSE 1986] |
| A. | \[\frac{1}{3}(x+\log x)+c\] |
| B. | \[\frac{1}{3}{{(x+\log x)}^{2}}+c\] |
| C. | \[\frac{1}{3}{{(x+\log x)}^{3}}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3987. |
\[\int_{{}}^{{}}{\frac{\cos 2x+x+1}{{{x}^{2}}+\sin 2x+2x}}\ dx=\] [AI CBSE 1980] |
| A. | \[\log ({{x}^{2}}+\sin 2x+2x)+c\] |
| B. | \[-\log ({{x}^{2}}+\sin 2x+2x)+c\] |
| C. | \[\frac{1}{2}\log ({{x}^{2}}+\sin 2x+2x)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3988. |
\[\int_{{}}^{{}}{{{a}^{3x+3}}dx}=\] [Roorkee 1977] |
| A. | \[\frac{{{a}^{3x+3}}}{\log a}+c\] |
| B. | \[\frac{{{a}^{3x+3}}}{3\log a}+c\] |
| C. | \[{{a}^{3x+3}}\log a+c\] |
| D. | \[3{{a}^{3x+3}}\log a+c\] |
| Answer» C. \[{{a}^{3x+3}}\log a+c\] | |
| 3989. |
\[\int_{{}}^{{}}{{{\sin }^{3}}x\ .\ \cos x\ dx=}\] [SCRA 1996] |
| A. | \[\frac{{{\sin }^{4}}x{{\cos }^{2}}x}{8}+c\] |
| B. | \[\frac{{{\sin }^{4}}x}{4}+c\] |
| C. | \[\frac{{{\sin }^{2}}x}{2}+c\] |
| D. | \[4{{\sin }^{4}}x+c\] |
| Answer» C. \[\frac{{{\sin }^{2}}x}{2}+c\] | |
| 3990. |
\[\int_{{}}^{{}}{\frac{x\ dx}{1-x\cot x}}=\] |
| A. | \[\log (\cos x-x\sin x)+c\] |
| B. | \[\log (x\sin x-\cos x)+c\] |
| C. | \[\log (\sin x-x\cos x)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3991. |
\[\int_{{}}^{{}}{{{e}^{-x}}\text{cose}{{\text{c}}^{2}}(2{{e}^{-x}}+5)}\ dx=\] [AISSE 1988] |
| A. | \[\frac{1}{2}\cot (2{{e}^{-x}}+5)+c\] |
| B. | \[-\frac{1}{2}\cot (2{{e}^{-x}}+5)+c\] |
| C. | \[2\cot (2{{e}^{-x}}+5)+c\] |
| D. | \[-2\cot (2{{e}^{-x}}+5)+c\] |
| Answer» B. \[-\frac{1}{2}\cot (2{{e}^{-x}}+5)+c\] | |
| 3992. |
\[\int_{{}}^{{}}{{{\sec }^{4}}x\tan x\ dx=}\] [AI CBSE 1980, 81; SCRA 1996] |
| A. | \[\frac{1}{4}{{\sec }^{4}}x+c\] |
| B. | \[4{{\sec }^{4}}x+c\] |
| C. | \[\frac{{{\sec }^{3}}x}{3}+c\] |
| D. | \[3{{\sec }^{3}}x+c\] |
| Answer» B. \[4{{\sec }^{4}}x+c\] | |
| 3993. |
\[\int_{{}}^{{}}{2x{{\cos }^{3}}{{x}^{2}}\sin {{x}^{2}}dx=}\] |
| A. | \[-\frac{1}{4}{{\cos }^{4}}{{x}^{2}}+c\] |
| B. | \[\frac{1}{4}{{\cos }^{4}}{{x}^{2}}+c\] |
| C. | \[{{\cos }^{4}}{{x}^{2}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{4}{{\cos }^{4}}{{x}^{2}}+c\] | |
| 3994. |
\[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{1-{{x}^{8}}}}dx=}\] |
| A. | \[\frac{1}{2}{{\sin }^{-1}}({{x}^{4}})+c\] |
| B. | \[\frac{1}{3}{{\sin }^{-1}}({{x}^{4}})+c\] |
| C. | \[\frac{1}{4}{{\sin }^{-1}}({{x}^{4}})+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 3995. |
\[\int_{{}}^{{}}{\frac{{{a}^{\sqrt{x}}}}{\sqrt{x}}dx=}\] [Roorkee 1990; MP PET 2001] |
| A. | \[2{{a}^{\sqrt{x}}}{{\log }_{e}}a+c\] |
| B. | \[2{{a}^{\sqrt{x}}}{{\log }_{a}}e+c\] |
| C. | \[2{{a}^{\sqrt{x}}}{{\log }_{10}}a+c\] |
| D. | \[2{{a}^{\sqrt{x}}}{{\log }_{a}}10+c\] |
| Answer» C. \[2{{a}^{\sqrt{x}}}{{\log }_{10}}a+c\] | |
| 3996. |
\[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x\ dx}{\sqrt{{{\tan }^{2}}x+4}}=}\] |
| A. | \[\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\] |
| B. | \[\frac{1}{2}\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\] |
| C. | \[\log \left[ \frac{1}{2}\tan x+\frac{1}{2}\sqrt{{{\tan }^{2}}x+4} \right]+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\] | |
| 3997. |
\[\int_{{}}^{{}}{\frac{dx}{x\log x\log (\log x)}=}\] |
| A. | \[2\log (\log x)+c\] |
| B. | \[\log [\log (\log x)]+c\] |
| C. | \[\log (x\log x)+c\] |
| D. | None of these |
| Answer» C. \[\log (x\log x)+c\] | |
| 3998. |
\[\int_{{}}^{{}}{\frac{1}{x}{{\sec }^{2}}(\log x)dx=}\] |
| A. | \[\tan (\log x)+c\] |
| B. | \[\log (\sec x)+c\] |
| C. | \[\log (\tan x)+c\] |
| D. | \[\sec (\log x)\ .\ \tan (\log x)+c\] |
| Answer» B. \[\log (\sec x)+c\] | |
| 3999. |
\[\int_{{}}^{{}}{\frac{1}{{{x}^{3}}}{{[\log {{x}^{x}}]}^{2}}\ dx=}\] |
| A. | \[\frac{{{x}^{3}}}{3}(\log x)+x+c\] |
| B. | \[\frac{1}{3}{{(\log x)}^{3}}+c\] |
| C. | \[3\log (\log x)+c\] |
| D. | None of these |
| Answer» C. \[3\log (\log x)+c\] | |
| 4000. |
\[\int_{{}}^{{}}{\frac{1}{\log a}({{a}^{x}}\cos {{a}^{x}})dx=}\] |
| A. | \[\sin {{a}^{x}}+c\] |
| B. | \[{{a}^{x}}\sin {{a}^{x}}+c\] |
| C. | \[\frac{1}{{{(\log a)}^{2}}}\sin {{a}^{x}}+c\] |
| D. | \[\log \sin {{a}^{x}}+c\] |
| Answer» D. \[\log \sin {{a}^{x}}+c\] | |