MCQOPTIONS
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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.
| 4001. |
To find the value of \[\int_{{}}^{{}}{\frac{dx}{x\sqrt{2ax-{{x}^{2}}}}}\], the suitable substitution is |
| A. | \[x=a\cos t\] |
| B. | \[x=2a\cos t\] |
| C. | \[x=2at\] |
| D. | \[x=2a{{\sin }^{2}}t\] |
| Answer» E. | |
| 4002. |
\[\int_{{}}^{{}}{\frac{{{e}^{x}}\ dx}{\sqrt{1-{{e}^{2x}}}}=}\] |
| A. | \[{{\cos }^{-1}}({{e}^{x}})+c\] |
| B. | \[-{{\cos }^{-1}}({{e}^{x}})+c\] |
| C. | \[{{\cos }^{-1}}({{e}^{2x}})+c\] |
| D. | \[\sqrt{1-{{e}^{2x}}}+c\] |
| Answer» C. \[{{\cos }^{-1}}({{e}^{2x}})+c\] | |
| 4003. |
\[\int_{{}}^{{}}{\tan x}{{\sec }^{2}}x\sqrt{1-{{\tan }^{2}}x}\ dx=\] |
| A. | \[-\frac{1}{3}{{(1-{{\tan }^{2}}x)}^{3/2}}+c\] |
| B. | \[\frac{1}{3}{{(1-{{\tan }^{2}}x)}^{3/2}}+c\] |
| C. | \[-\frac{2}{3}{{(1-{{\tan }^{2}}x)}^{2/3}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{3}{{(1-{{\tan }^{2}}x)}^{3/2}}+c\] | |
| 4004. |
For which of the following functions, the substitution \[{{x}^{2}}=t\]is applicable |
| A. | \[\int_{{}}^{{}}{{{x}^{6}}{{\tan }^{-1}}{{x}^{3}}}\ dx\] |
| B. | \[\int_{{}}^{{}}{{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)\ dx}\] |
| C. | \[\int_{{}}^{{}}{{{x}^{3}}\cos {{x}^{2}}\ dx}\] |
| D. | None of these |
| Answer» D. None of these | |
| 4005. |
\[\int_{{}}^{{}}{\frac{f'(x)}{{{[f(x)]}^{2}}}}\ dx=\] |
| A. | \[-{{[f(x)]}^{-1}}+c\] |
| B. | \[\log [f(x)]+c\] |
| C. | \[{{e}^{f(x)}}+c\] |
| D. | None of these |
| Answer» B. \[\log [f(x)]+c\] | |
| 4006. |
\[\int_{{}}^{{}}{\frac{dx}{x\sqrt{1-{{(\log x)}^{2}}}}=}\] |
| A. | \[{{\cos }^{-1}}(\log x)+c\] |
| B. | \[x\log (1-{{x}^{2}})+c\] |
| C. | \[{{\sin }^{-1}}(\log x)+c\] |
| D. | \[\frac{1}{2}{{\cos }^{-1}}(\log x)+c\] |
| Answer» D. \[\frac{1}{2}{{\cos }^{-1}}(\log x)+c\] | |
| 4007. |
\[\int_{{}}^{{}}{\frac{dx}{{{e}^{-2x}}{{({{e}^{2x}}+1)}^{2}}}=}\] |
| A. | \[\frac{-1}{2({{e}^{2x}}+1)}+c\] |
| B. | \[\frac{1}{2({{e}^{2x}}+1)}+c\] |
| C. | \[\frac{1}{{{e}^{2x}}+1}+c\] |
| D. | \[\frac{-1}{{{e}^{2x}}+1}+c\] |
| Answer» B. \[\frac{1}{2({{e}^{2x}}+1)}+c\] | |
| 4008. |
\[\int_{{}}^{{}}{{{e}^{x}}{{\tan }^{2}}({{e}^{x}})dx=}\] |
| A. | \[\tan ({{e}^{x}})-x+c\] |
| B. | \[{{e}^{x}}(\tan {{e}^{x}}-1)+c\] |
| C. | \[\sec ({{e}^{x}})+c\] |
| D. | \[\tan ({{e}^{x}})-{{e}^{x}}+c\] |
| Answer» E. | |
| 4009. |
\[\int_{{}}^{{}}{\frac{\sec x\ dx}{\sqrt{\cos 2x}}}=\] |
| A. | \[{{\sin }^{-1}}(\tan x)\] |
| B. | \[\tan x\] |
| C. | \[{{\cos }^{-1}}(\tan x)\] |
| D. | \[\frac{\sin x}{\sqrt{\cos x}}\] |
| Answer» B. \[\tan x\] | |
| 4010. |
\[\int_{{}}^{{}}{\frac{1}{\sqrt{x}}}\sin \sqrt{x}\ dx=\] [MP PET 1989] |
| A. | \[-\frac{1}{2}\cos \sqrt{x}+c\] |
| B. | \[-2\cos \sqrt{x}+c\] |
| C. | \[\frac{1}{2}\cos \sqrt{x}+c\] |
| D. | \[2\cos \sqrt{x}+c\] |
| Answer» C. \[\frac{1}{2}\cos \sqrt{x}+c\] | |
| 4011. |
\[\int_{{}}^{{}}{\frac{\text{cose}{{\text{c}}^{2}}x}{1+\cot x}dx=}\] [MNR 1973] |
| A. | \[\log (1+\cot x)+c\] |
| B. | \[-\log (1+\cot x)+c\] |
| C. | \[\frac{1}{2{{(1+\cot x)}^{2}}}+c\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2{{(1+\cot x)}^{2}}}+c\] | |
| 4012. |
To evaluate \[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{(1+\tan x)(2+\tan x)}\ dx}\], the most suitable substitution is |
| A. | \[1+\tan x=t\] |
| B. | \[2+\tan x=t\] |
| C. | \[\tan x=t\] |
| D. | None of these |
| Answer» D. None of these | |
| 4013. |
\[\int_{{}}^{{}}{\frac{1}{{{\cos }^{-1}}x.\sqrt{1-{{x}^{2}}}}dx=}\] |
| A. | \[\log ({{\cos }^{-1}}x)+c\] |
| B. | \[-\log ({{\cos }^{-1}}x)+c\] |
| C. | \[-\frac{1}{2{{({{\cos }^{-1}}x)}^{2}}}+c\] |
| D. | None of these |
| Answer» C. \[-\frac{1}{2{{({{\cos }^{-1}}x)}^{2}}}+c\] | |
| 4014. |
To evaluate \[\int_{{}}^{{}}{{{x}^{3}}{{e}^{3{{x}^{2}}+5}}}dx\], the simplest way is to |
| A. | Substitute \[{{x}^{2}}=t\] |
| B. | Substitute \[(3{{x}^{2}}+5)=t\] |
| C. | Integrate by parts |
| D. | None of these |
| Answer» C. Integrate by parts | |
| 4015. |
\[\int_{{}}^{{}}{\sec x{{\tan }^{3}}x\ dx=}\] |
| A. | \[\frac{1}{3}{{\sec }^{3}}x-\sec x+c\] |
| B. | \[{{\sec }^{3}}x-\sec x+c\] |
| C. | \[\frac{1}{3}{{\sec }^{3}}x+\sec x+c\] |
| D. | None of these |
| Answer» B. \[{{\sec }^{3}}x-\sec x+c\] | |
| 4016. |
\[\int_{{}}^{{}}{{{\cos }^{5}}x\ dx=}\] |
| A. | \[\sin x-\frac{2}{3}{{\sin }^{3}}x+\frac{1}{5}{{\sin }^{5}}x+c\] |
| B. | \[\sin x+\frac{2}{3}{{\sin }^{3}}x+\frac{1}{5}{{\sin }^{5}}x+c\] |
| C. | \[\sin x-\frac{2}{3}{{\sin }^{3}}x-\frac{1}{5}{{\sin }^{5}}x+c\] |
| D. | None of these |
| Answer» B. \[\sin x+\frac{2}{3}{{\sin }^{3}}x+\frac{1}{5}{{\sin }^{5}}x+c\] | |
| 4017. |
\[\int_{{}}^{{}}{{{\sec }^{2/3}}x\,\text{cose}{{\text{c}}^{4/3}}x\ dx=}\] |
| A. | \[-3{{(\tan x)}^{1/3}}+c\] |
| B. | \[-3{{(\tan x)}^{-1/3}}+c\] |
| C. | \[3{{(\tan x)}^{-1/3}}+c\] |
| D. | \[{{(\tan x)}^{-1/3}}+c\] |
| Answer» C. \[3{{(\tan x)}^{-1/3}}+c\] | |
| 4018. |
\[\int_{{}}^{{}}{\cos x\sqrt{4-{{\sin }^{2}}x}}\ dx=\] |
| A. | \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}-2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] |
| B. | \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] |
| C. | \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\] | |
| 4019. |
\[\int_{{}}^{{}}{{{x}^{2}}{{(3)}^{{{x}^{3}}+1}}dx=}\] |
| A. | \[{{(3)}^{{{x}^{3}}}}+c\] |
| B. | \[\frac{{{(3)}^{{{x}^{3}}}}}{\log 3}+c\] |
| C. | \[\log 3{{(3)}^{{{x}^{3}}}}+c\] |
| D. | None of these |
| Answer» C. \[\log 3{{(3)}^{{{x}^{3}}}}+c\] | |
| 4020. |
\[\int_{{}}^{{}}{\frac{3{{x}^{2}}}{\sqrt{9-16{{x}^{6}}}}}\ dx=\] |
| A. | \[\frac{1}{4}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\] |
| B. | \[\frac{1}{3}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\] |
| C. | \[\frac{1}{4}{{\sin }^{-1}}{{x}^{3}}+c\] |
| D. | \[\frac{1}{3}{{\sin }^{-1}}{{x}^{3}}+c\] |
| Answer» B. \[\frac{1}{3}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\] | |
| 4021. |
To find the value of \[\int_{{}}^{{}}{\frac{1+\log x}{x}\text{ }}dx\], the proper substitution is [MP PET 1988] |
| A. | \[\log x=t\] |
| B. | \[1+\log x=t\] |
| C. | \[\frac{1}{x}=t\] |
| D. | None of these |
| Answer» C. \[\frac{1}{x}=t\] | |
| 4022. |
\[\int_{{}}^{{}}{\frac{1}{\sqrt{1-{{e}^{2x}}}}\ dx=}\] [MP PET 1993, 2002; RPET 1999] |
| A. | \[x-\log [1+\sqrt{1-{{e}^{2x}}}]+c\] |
| B. | \[x+\log [1+\sqrt{1-{{e}^{2x}}}]+c\] |
| C. | \[\log [1+\sqrt{1-{{e}^{2x}}}]-x+c\] |
| D. | None of these |
| Answer» B. \[x+\log [1+\sqrt{1-{{e}^{2x}}}]+c\] | |
| 4023. |
\[\int_{{}}^{{}}{\frac{{{e}^{-x}}}{1+{{e}^{x}}}\ dx=}\] |
| A. | \[\log (1+{{e}^{x}})-x-{{e}^{-x}}+c\] |
| B. | \[\log (1+{{e}^{x}})+x-{{e}^{-x}}+c\] |
| C. | \[\log (1+{{e}^{x}})-x+{{e}^{-x}}+c\] |
| D. | \[\log (1+{{e}^{x}})+x+{{e}^{-x}}+c\] |
| Answer» B. \[\log (1+{{e}^{x}})+x-{{e}^{-x}}+c\] | |
| 4024. |
\[\int_{{}}^{{}}{\frac{x}{1+{{x}^{4}}}\ dx=}\] [IIT 1978; UPSEAT 2002] |
| A. | \[\frac{1}{2}{{\cot }^{-1}}{{x}^{2}}+c\] |
| B. | \[\frac{1}{2}{{\tan }^{-1}}{{x}^{2}}+c\] |
| C. | \[{{\cot }^{-1}}{{x}^{2}}+c\] |
| D. | \[{{\tan }^{-1}}{{x}^{2}}+c\] |
| Answer» C. \[{{\cot }^{-1}}{{x}^{2}}+c\] | |
| 4025. |
\[\int_{{}}^{{}}{\tan (3x-5)\sec (3x-5)\ dx=}\] [MP PET 1988] |
| A. | \[\sec (3x-5)+c\] |
| B. | \[\frac{1}{3}\sec (3x-5)+c\] |
| C. | \[\tan (3x-5)+c\] |
| D. | None of these |
| Answer» C. \[\tan (3x-5)+c\] | |
| 4026. |
\[\int_{{}}^{{}}{{{\cos }^{3}}x\ {{e}^{\log (\sin x)}}}\ dx\] is equal to |
| A. | \[-\frac{{{\sin }^{4}}x}{4}+c\] |
| B. | \[-\frac{{{\cos }^{4}}x}{4}+c\] |
| C. | \[\frac{{{e}^{\sin x}}}{4}+c\] |
| D. | None of these |
| Answer» C. \[\frac{{{e}^{\sin x}}}{4}+c\] | |
| 4027. |
\[\int_{{}}^{{}}{\frac{\tan (\log x)}{x}\ dx=}\] |
| A. | \[\log \cos (\log x)+c\] |
| B. | \[\log \sin (\log x)+c\] |
| C. | \[\log \sec (\log x)+c\] |
| D. | \[\log \text{cosec}(\log x)+c\] |
| Answer» D. \[\log \text{cosec}(\log x)+c\] | |
| 4028. |
\[\int_{{}}^{{}}{\frac{1}{{{({{e}^{x}}+{{e}^{-x}})}^{2}}}\ dx=}\] |
| A. | \[-\frac{1}{2({{e}^{2x}}+1)}+c\] |
| B. | \[\frac{1}{2({{e}^{2x}}+1)}+c\] |
| C. | \[-\frac{1}{{{e}^{2x}}+1}\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2({{e}^{2x}}+1)}+c\] | |
| 4029. |
\[\int_{{}}^{{}}{\frac{dx}{x+x\log x}=}\] [MP PET 1993; Roorkee 1977] |
| A. | \[\log (1+\log x)\] |
| B. | \[\log \log (1+\log x)\] |
| C. | \[\log x+\log (\log x)\] |
| D. | None of these |
| Answer» B. \[\log \log (1+\log x)\] | |
| 4030. |
\[\int_{{}}^{{}}{\frac{10{{x}^{9}}+{{10}^{x}}{{\log }_{e}}10}{{{10}^{x}}+{{x}^{10}}}}\ dx=\] [MNR 1979] |
| A. | \[-\frac{1}{2}\frac{1}{{{({{10}^{x}}+{{x}^{10}})}^{2}}}+c\] |
| B. | \[\log ({{10}^{x}}+{{x}^{10}})+c\] |
| C. | \[\frac{1}{2}\frac{1}{{{({{10}^{x}}+{{x}^{10}})}^{2}}}+c\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}\frac{1}{{{({{10}^{x}}+{{x}^{10}})}^{2}}}+c\] | |
| 4031. |
\[\int_{{}}^{{}}{\frac{\cos \text{ec}x}{\log \tan \frac{x}{2}}\ dx=}\] |
| A. | \[\log \left( \log \tan \frac{x}{2} \right)+c\] |
| B. | \[2\log \left( \log \tan \frac{x}{2} \right)+c\] |
| C. | \[\frac{1}{2}\log \left( \log \tan \frac{x}{2} \right)+c\] |
| D. | None of these |
| Answer» B. \[2\log \left( \log \tan \frac{x}{2} \right)+c\] | |
| 4032. |
\[\int_{{}}^{{}}{\frac{{{e}^{2x}}-1}{{{e}^{2x}}+1}}\ dx=\] [MP PET 1987] |
| A. | \[\frac{{{e}^{2x}}-1}{{{e}^{2x}}+1}+c\] |
| B. | \[\log ({{e}^{2x}}+1)-x+c\] |
| C. | \[\log ({{e}^{2x}}+1)+c\] |
| D. | None of these |
| Answer» C. \[\log ({{e}^{2x}}+1)+c\] | |
| 4033. |
\[\int_{{}}^{{}}{\frac{1}{x\sqrt{1+\log x}}\ dx=}\] [Roorkee 1977] |
| A. | \[\frac{2}{3}{{(1+\log x)}^{3/2}}+c\] |
| B. | \[{{(1+\log x)}^{3/2}}+c\] |
| C. | \[2\sqrt{1+\log x}+c\] |
| D. | \[\sqrt{1+\log x}+c\] |
| Answer» D. \[\sqrt{1+\log x}+c\] | |
| 4034. |
\[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x}{1+\tan x}\ dx=}\] [MP PET 1987] |
| A. | \[\log (\cos x+\sin x)+c\] |
| B. | \[\log ({{\sec }^{2}}x)+c\] |
| C. | \[\log (1+\tan x)+c\] |
| D. | \[-\frac{1}{{{(1+\tan x)}^{2}}}+c\] |
| Answer» D. \[-\frac{1}{{{(1+\tan x)}^{2}}}+c\] | |
| 4035. |
\[\int_{{}}^{{}}{\frac{\sin 2x}{{{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x}}\ dx=\] [Roorkee 1977] |
| A. | \[\frac{1}{{{b}^{2}}}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] |
| B. | \[\frac{1}{b}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] |
| C. | \[\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] |
| D. | \[{{b}^{2}}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] |
| Answer» B. \[\frac{1}{b}\log ({{a}^{2}}+{{b}^{2}}{{\sin }^{2}}x)+c\] | |
| 4036. |
\[\int_{{}}^{{}}{\frac{\sqrt{\tan x}}{\sin x\cos x}}\ dx=\] [Bihar CEE 1974; MP PET 2002; Kerala (Engg.) 2002] |
| A. | \[2\sqrt{\sec x}+c\] |
| B. | \[2\sqrt{\tan x}+c\] |
| C. | \[\frac{2}{\sqrt{\tan x}}+c\] |
| D. | \[\frac{2}{\sqrt{\sec x}}+c\] |
| Answer» C. \[\frac{2}{\sqrt{\tan x}}+c\] | |
| 4037. |
\[\int_{{}}^{{}}{\frac{{{a}^{x}}}{\sqrt{1-{{a}^{2x}}}}dx=}\] [MNR 1983, 87] |
| A. | \[\frac{1}{\log a}{{\sin }^{-1}}{{a}^{x}}+c\] |
| B. | \[{{\sin }^{-1}}{{a}^{x}}+c\] |
| C. | \[\frac{1}{\log a}{{\cos }^{-1}}{{a}^{x}}+c\] |
| D. | \[{{\cos }^{-1}}{{a}^{x}}+c\] |
| Answer» B. \[{{\sin }^{-1}}{{a}^{x}}+c\] | |
| 4038. |
\[\int_{{}}^{{}}{\frac{1}{\sqrt{x}}{{\tan }^{4}}\sqrt{x}}{{\sec }^{2}}\sqrt{x}\ dx=\] |
| A. | \[2{{\tan }^{5}}\sqrt{x}+c\] |
| B. | \[\frac{1}{5}{{\tan }^{5}}\sqrt{x}+c\] |
| C. | \[\frac{2}{5}{{\tan }^{5}}\sqrt{x}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4039. |
\[\int_{{}}^{{}}{\frac{{{e}^{\sqrt{x}}}\cos {{e}^{\sqrt{x}}}}{\sqrt{x}}dx}=\] |
| A. | \[2\sin {{e}^{\sqrt{x}}}\] |
| B. | \[\sin {{e}^{\sqrt{x}}}\] |
| C. | \[2\cos {{e}^{\sqrt{x}}}\] |
| D. | \[-2\sin {{e}^{\sqrt{x}}}\] |
| Answer» B. \[\sin {{e}^{\sqrt{x}}}\] | |
| 4040. |
\[\int_{{}}^{{}}{\frac{{{e}^{{{\tan }^{-1}}x}}}{1+{{x}^{2}}}dx=}\] [MP PET 1987] |
| A. | \[\log (1+{{x}^{2}})+c\] |
| B. | \[\log {{e}^{{{\tan }^{-1}}x}}+c\] |
| C. | \[{{e}^{{{\tan }^{-1}}x}}+c\] |
| D. | \[{{\tan }^{-1}}{{e}^{{{\tan }^{-1}}x}}+c\] |
| Answer» D. \[{{\tan }^{-1}}{{e}^{{{\tan }^{-1}}x}}+c\] | |
| 4041. |
\[\int_{{}}^{{}}{\frac{\sin x\cos x}{a{{\cos }^{2}}x+b{{\sin }^{2}}x}dx=}\] [AI CBSE 1988, 89] |
| A. | \[\frac{1}{2(b-a)}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] |
| B. | \[\frac{1}{b-a}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] |
| C. | \[\frac{1}{2}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{b-a}\log (a{{\cos }^{2}}x+b{{\sin }^{2}}x)+c\] | |
| 4042. |
\[\int_{{}}^{{}}{\frac{x+1}{\sqrt{1+{{x}^{2}}}}dx}=\] [MP PET 1991] |
| A. | \[\sqrt{1+{{x}^{2}}}+{{\tan }^{-1}}x+c\] |
| B. | \[\sqrt{1+{{x}^{2}}}-\log \{x+\sqrt{1+{{x}^{2}}}\}+c\] |
| C. | \[\sqrt{1+{{x}^{2}}}+\log \{x+\sqrt{1+{{x}^{2}}}\}+c\] |
| D. | \[\sqrt{1+{{x}^{2}}}+\log (\sec x+\tan x)+c\] |
| Answer» D. \[\sqrt{1+{{x}^{2}}}+\log (\sec x+\tan x)+c\] | |
| 4043. |
\[\int_{{}}^{{}}{\frac{{{e}^{x}}(x+1)}{{{\cos }^{2}}(x{{e}^{x}})}dx=}\] [Roorkee 1979; MP PET 1995; Pb. CET 2001] |
| A. | \[\tan (x{{e}^{x}})+c\] |
| B. | \[\sec (x{{e}^{x}})\tan (x{{e}^{x}})+c\] |
| C. | \[-\tan (x{{e}^{x}})+c\] |
| D. | None of these |
| Answer» B. \[\sec (x{{e}^{x}})\tan (x{{e}^{x}})+c\] | |
| 4044. |
\[\int_{{}}^{{}}{x\sqrt{1+{{x}^{2}}}}\ dx=\] [MP PET 1989] |
| A. | \[\frac{1+2{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}+c\] |
| B. | \[\sqrt{1+{{x}^{2}}}+c\] |
| C. | \[3{{(1+{{x}^{2}})}^{3/2}}+c\] |
| D. | \[\frac{1}{3}{{(1+{{x}^{2}})}^{3/2}}+c\] |
| Answer» E. | |
| 4045. |
\[\int_{{}}^{{}}{\frac{x-2}{x(2\log x-x)}dx}=\] |
| A. | \[\log (2\log x-x)+c\] |
| B. | \[\log \left( \frac{1}{2\log x-x} \right)+c\] |
| C. | \[\log (x-2\log x)+c\] |
| D. | \[\log \left( \frac{1}{x-2\log x} \right)+c\] |
| Answer» C. \[\log (x-2\log x)+c\] | |
| 4046. |
\[\int_{{}}^{{}}{\frac{\sin 2x}{{{\sin }^{4}}x+{{\cos }^{4}}x}dx=}\] [RPET 1995] |
| A. | \[{{\cot }^{-1}}({{\tan }^{2}}x)+c\] |
| B. | \[{{\tan }^{-1}}({{\tan }^{2}}x)+c\] |
| C. | \[{{\cot }^{-1}}({{\cot }^{2}}x)+c\] |
| D. | \[{{\tan }^{-1}}({{\cot }^{2}}x)+c\] |
| Answer» C. \[{{\cot }^{-1}}({{\cot }^{2}}x)+c\] | |
| 4047. |
\[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}+{{e}^{-x}}}=}\] [Bihar CEE 1976; MNR 1974] |
| A. | \[{{\tan }^{-1}}({{e}^{-x}})\] |
| B. | \[{{\tan }^{-1}}({{e}^{x}})\] |
| C. | \[\log ({{e}^{x}}-{{e}^{-x}})\] |
| D. | \[\log ({{e}^{x}}+{{e}^{-x}})\] |
| Answer» C. \[\log ({{e}^{x}}-{{e}^{-x}})\] | |
| 4048. |
\[\int_{{}}^{{}}{{{x}^{2}}\sec {{x}^{3}}\ dx}=\] [MNR 1986; Roorkee 1975] |
| A. | \[\log (\sec {{x}^{3}}+\tan {{x}^{3}})\] |
| B. | \[3(\sec {{x}^{3}}+\tan {{x}^{3}})\] |
| C. | \[\frac{1}{3}\log (\sec {{x}^{3}}+\tan {{x}^{3}})\] |
| D. | None of these |
| Answer» D. None of these | |
| 4049. |
\[\int_{{}}^{{}}{\frac{dx}{{{e}^{x}}-1}=}\] [MP PET 1989] |
| A. | \[\ln (1-{{e}^{-x}})+c\] |
| B. | \[-\ln (1-{{e}^{-x}})+c\] |
| C. | \[\ln ({{e}^{x}}-1)+c\] |
| D. | None of these |
| Answer» B. \[-\ln (1-{{e}^{-x}})+c\] | |
| 4050. |
\[\int_{{}}^{{}}{{{\sec }^{p}}x\tan x\ dx=}\] |
| A. | \[\frac{{{\sec }^{p+1}}x}{p+1}+c\] |
| B. | \[\frac{{{\sec }^{p}}x}{p}+c\] |
| C. | \[\frac{{{\tan }^{p+1}}x}{p+1}+c\] |
| D. | \[\frac{{{\tan }^{p}}x}{p}+c\] |
| Answer» C. \[\frac{{{\tan }^{p+1}}x}{p+1}+c\] | |