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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.
| 4051. |
\[\int_{{}}^{{}}{\frac{{{(1+\log x)}^{2}}}{x}}\ dx=\] [Roorkee 1977] |
| A. | \[{{(1+\log x)}^{3}}+c\] |
| B. | \[3{{(1+\log x)}^{3}}+c\] |
| C. | \[\frac{1}{3}{{(1+\log x)}^{3}}+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4052. |
\[\int{\frac{(\sin \theta +\cos \theta )}{\sqrt{\sin 2\theta }}}d\theta =\] [Kerala (Engg.) 2005] |
| A. | \[\log \left| \cos \theta -\sin \theta +\sqrt{\sin 2\theta } \right|\] |
| B. | \[\log \left| \sin \theta -\cos \theta )+\sqrt{\sin 2\theta } \right|\] |
| C. | \[{{\sin }^{-1}}(\sin \theta -\cos \theta )+c\] |
| D. | \[{{\sin }^{-1}}(\sin \theta +\cos \theta )+c\] |
| E. | \[{{\sin }^{-1}}(\cos \theta -\sin \theta )+c\] |
| Answer» D. \[{{\sin }^{-1}}(\sin \theta +\cos \theta )+c\] | |
| 4053. |
\[\int_{{}}^{{}}{\frac{3{{x}^{2}}}{{{x}^{6}}+1}dx=}\] [MNR 1981; MP PET 1988; RPET 1995] |
| A. | \[\log ({{x}^{6}}+1)+c\] |
| B. | \[{{\tan }^{-1}}({{x}^{3}})+c\] |
| C. | \[3{{\tan }^{-1}}({{x}^{3}})+c\] |
| D. | \[3{{\tan }^{-1}}\left( \frac{{{x}^{3}}}{3} \right)+c\] |
| Answer» C. \[3{{\tan }^{-1}}({{x}^{3}})+c\] | |
| 4054. |
\[\int{{{\cos }^{-3/7}}}x{{\sin }^{-11/7}}x\,\,dx=\] [Kerala (Engg.) 2005] |
| A. | \[\log |{{\sin }^{4/7}}x|+c\] |
| B. | \[\frac{4}{7}{{\tan }^{4/7}}x+c\] |
| C. | \[\frac{-7}{4}{{\tan }^{-4/7}}x+c\] |
| D. | \[\log |{{\cos }^{3/7}}x|+c\] |
| E. | \[\frac{7}{4}{{\tan }^{-4/7}}x+C\] |
| Answer» D. \[\log |{{\cos }^{3/7}}x|+c\] | |
| 4055. |
\[\int{\frac{dx}{\sin (x-a)\sin (x-b)}}\] is [Kerala (Engg.) 2005] |
| A. | \[\frac{1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\] |
| B. | \[\frac{-1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\] |
| C. | \[\log \sin (x-a)\sin (x-b)+c\] |
| D. | \[\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|\] |
| E. | \[\frac{1}{\sin (x-a)}\log \sin (x-a)\sin (x-b)+c\] |
| Answer» B. \[\frac{-1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c\] | |
| 4056. |
\[\int{\sqrt{{{e}^{x}}-1}}dx=\] [Kerala (Engg.) 2005] |
| A. | \[2\left[ \sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1} \right]+c\] |
| B. | \[\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c\] |
| C. | \[\sqrt{{{e}^{x}}-1}+{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c\] |
| D. | \[2\left[ \sqrt{{{e}^{x}}-1}+{{\tan }^{-1}}\sqrt{{{e}^{x}}-1} \right]+c\] |
| E. | \[2\left[ \sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}+1} \right]+c\] |
| Answer» B. \[\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c\] | |
| 4057. |
If \[\int{\frac{\cos 4x+1}{\cos x-\tan x}}dx=k\,\,\cos 4x+c\] then [DCE 2005] |
| A. | \[k=-1/2\] |
| B. | \[k=-1/8\] |
| C. | \[k=-1/4\] |
| D. | None of these |
| Answer» C. \[k=-1/4\] | |
| 4058. |
Let \[f(x)=\int{\frac{{{x}^{2}}dx}{(1+{{x}^{2}})\,\left( 1+\sqrt{1+{{x}^{2}}} \right)}}\]and \[f(0)=0\], then the value of \[f(1)\] be [AMU 2005] |
| A. | \[\log (1+\sqrt{2})\] |
| B. | \[\log (1+\sqrt{2})-\frac{\pi }{4}\] |
| C. | \[\log (1+\sqrt{2})+\frac{\pi }{2}\] |
| D. | None of these |
| Answer» C. \[\log (1+\sqrt{2})+\frac{\pi }{2}\] | |
| 4059. |
If \[\int{\frac{1}{x+{{x}^{5}}}dx=f(x)+c}\], then the value of \[\int{\frac{{{x}^{4}}}{x+{{x}^{5}}}dx}\] is [DCE 2005] |
| A. | \[\log x-f(x)+c\] |
| B. | \[f(x)+\log x+c\] |
| C. | \[f(x)-\log x+c\] |
| D. | None of these |
| Answer» B. \[f(x)+\log x+c\] | |
| 4060. |
\[\int_{{}}^{{}}{\frac{\sin 2xdx}{1+{{\cos }^{2}}x}}=\] [Karnataka CET 2005] |
| A. | \[\frac{1}{2}\log (1+{{\cos }^{2}}x)+c\] |
| B. | \[2\log (1+{{\cos }^{2}}x)+c\] |
| C. | \[\frac{1}{2}\log (1+\cos 2x)+c\] |
| D. | \[-\log (1+{{\cos }^{2}}x)+c\] |
| Answer» E. | |
| 4061. |
\[{{\int{\left\{ \frac{(\log x-1)}{1+{{(\log x)}^{2}}} \right\}}}^{2}}dx\] is equal to [AIEEE 2005] |
| A. | \[\frac{x{{e}^{x}}}{1+{{x}^{2}}}+c\] |
| B. | \[\frac{x}{{{(\log x)}^{2}}+1}+C\] |
| C. | \[\frac{\log x}{{{(\log x)}^{2}}+1}+c\] |
| D. | \[\frac{x}{{{x}^{2}}+1}+c\] |
| Answer» C. \[\frac{\log x}{{{(\log x)}^{2}}+1}+c\] | |
| 4062. |
\[\int{\frac{\sin x\,\,dx}{3+4{{\cos }^{2}}x}=}\] [Karnataka CET 2000] |
| A. | \[\log (3+4{{\cos }^{2}}x)+c\] |
| B. | \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{\cos x}{\sqrt{3}} \right)+c\] |
| C. | \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\] |
| D. | \[\frac{1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\] |
| Answer» D. \[\frac{1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\] | |
| 4063. |
The value of \[\int_{{}}^{{}}{\frac{dx}{\sqrt{x}\,(x+9)}dx}\] is equal to [Pb. CET 2002] |
| A. | \[{{\tan }^{-1}}\sqrt{x}\] |
| B. | \[{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\] |
| C. | \[\frac{2}{3}{{\tan }^{-1}}\sqrt{x}\] |
| D. | \[\frac{2}{3}{{\tan }^{-1}}\left( \frac{\sqrt{x}}{3} \right)\] |
| Answer» E. | |
| 4064. |
\[\int_{{}}^{{}}{\frac{x-2}{{{x}^{2}}-4x+3}dx=}\] [MP PET 1987] |
| A. | \[\log \sqrt{{{x}^{2}}-4x+3}+c\] |
| B. | \[x\log (x-3)-2\log (x-2)+c\] |
| C. | \[\log [(x-3)(x-1)]\] |
| D. | None of these |
| Answer» B. \[x\log (x-3)-2\log (x-2)+c\] | |
| 4065. |
\[\int_{{}}^{{}}{\frac{x}{\sqrt{4-{{x}^{4}}}}dx}=\] [Roorkee 1976] |
| A. | \[{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\] |
| B. | \[\frac{1}{2}{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\] |
| C. | \[{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\] |
| D. | \[\frac{1}{2}{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\] |
| Answer» E. | |
| 4066. |
\[\int{\sqrt{\frac{1+x}{1-x}}\,\,dx=}\] [RPET 2002] |
| A. | \[-{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}\,+c\] |
| B. | \[{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}\,+c\] |
| C. | \[{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}\,+c\] |
| D. | \[-{{\sin }^{-1}}x-\sqrt{{{x}^{2}}-1}\,+c\] |
| Answer» D. \[-{{\sin }^{-1}}x-\sqrt{{{x}^{2}}-1}\,+c\] | |
| 4067. |
\[\int_{{}}^{{}}{\frac{\sin x}{\sin x-\cos x}}\ dx=\] [Roorkee 1988] |
| A. | \[\frac{1}{2}\log (\sin x-\cos x)+x+c\] |
| B. | \[\frac{1}{2}[\log (\sin x-\cos x)+x]+c\] |
| C. | \[\frac{1}{2}\log (\cos x-\sin x)+x+c\] |
| D. | \[\frac{1}{2}[\log (\cos x-\sin x)+x]+c\] |
| Answer» C. \[\frac{1}{2}\log (\cos x-\sin x)+x+c\] | |
| 4068. |
\[\int_{{}}^{{}}{\sqrt{\frac{1-x}{1+x}}}\ dx=\] [IIT 1971] |
| A. | \[{{\sin }^{-1}}x-\frac{1}{2}\sqrt{1-{{x}^{2}}}+c\] |
| B. | \[{{\sin }^{-1}}x+\frac{1}{2}\sqrt{1-{{x}^{2}}}+c\] |
| C. | \[{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] |
| D. | \[{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] |
| Answer» E. | |
| 4069. |
\[\int_{{}}^{{}}{\frac{\sqrt{x}}{1+x}dx=}\] |
| A. | \[\sqrt{x}-{{\tan }^{-1}}\sqrt{x}+c\] |
| B. | \[2(\sqrt{x}-{{\tan }^{-1}}\sqrt{x})+c\] |
| C. | \[2(\sqrt{x}+{{\tan }^{-1}}x)+c\] |
| D. | \[\sqrt{1+x}+c\] |
| Answer» C. \[2(\sqrt{x}+{{\tan }^{-1}}x)+c\] | |
| 4070. |
\[\int_{{}}^{{}}{\frac{{{({{\tan }^{-1}}x)}^{3}}}{1+{{x}^{2}}}\,dx=}\] [UPSEAT 2004] |
| A. | \[{{({{\tan }^{-1}}x)}^{4}}+c\] |
| B. | \[\frac{{{({{\tan }^{-1}}x)}^{4}}}{4}+c\] |
| C. | \[2{{\tan }^{-1}}x+c\] |
| D. | \[2{{({{\tan }^{-1}}x)}^{2}}+c\] |
| Answer» C. \[2{{\tan }^{-1}}x+c\] | |
| 4071. |
The value of \[\int_{{}}^{{}}{\frac{\sin x-\cos x}{\sin x+\cos x}\,dx}\] is [Pb. CET 2000] |
| A. | \[\frac{1}{\sin x+\cos x}+c\] |
| B. | \[\frac{1}{\sin x-\cos x}+c\] |
| C. | \[\log (\sin x+\cos x)+c\] |
| D. | \[\log \left( \frac{1}{\sin x+\cos x} \right)+c\] |
| Answer» E. | |
| 4072. |
The value of \[\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{e}^{x}}+1}}\,dx\] is [Pb. CET 2000] |
| A. | \[{{e}^{x}}+c\] |
| B. | \[({{e}^{x}}+1)+c\] |
| C. | \[\log ({{e}^{x}}+1)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4073. |
\[\int_{{}}^{{}}{\sec x\log (\sec x+\tan x)\ dx=}\] |
| A. | \[{{[\log (\sec x+\tan x)]}^{2}}+c\] |
| B. | \[\frac{1}{2}{{[\log (\sec x+\tan x)]}^{2}}+c\] |
| C. | \[{{\sec }^{2}}x+\tan x\sec x+c\] |
| D. | None of these |
| Answer» C. \[{{\sec }^{2}}x+\tan x\sec x+c\] | |
| 4074. |
. If \[\int{f(x)\,\,dx=g(x),}\] then \[\int{{{f}^{-1}}(x)}\,\,dx\] is equal to [MP PET 2003] |
| A. | \[{{g}^{-1}}(x)\] |
| B. | \[x{{f}^{-1}}(x)-g({{f}^{-1}}(x))\] |
| C. | \[x{{f}^{-1}}(x)-{{g}^{-1}}(x)\] |
| D. | \[{{f}^{-1}}(x)\] |
| Answer» C. \[x{{f}^{-1}}(x)-{{g}^{-1}}(x)\] | |
| 4075. |
\[\int{x{{e}^{{{x}^{2}}}}}dx=\] [RPET 2003] |
| A. | \[-\frac{{{e}^{{{x}^{2}}}}}{2}+c\] |
| B. | \[\frac{{{e}^{{{x}^{2}}}}}{2}+c\] |
| C. | \[\frac{{{e}^{x}}}{2}+c\] |
| D. | \[-\frac{{{e}^{x}}}{2}+c\] |
| Answer» C. \[\frac{{{e}^{x}}}{2}+c\] | |
| 4076. |
\[\int{\text{cose}{{\text{c}}^{4}}x\,dx}=\] [RPET 2002] |
| A. | \[\cot x+\frac{{{\cot }^{3}}x}{3}+c\] |
| B. | \[\tan x+\frac{{{\tan }^{3}}x}{3}+c\] |
| C. | \[-\cot x-\frac{{{\cot }^{3}}x}{3}+c\] |
| D. | \[-\tan x-\frac{{{\tan }^{3}}x}{3}+c\] |
| Answer» D. \[-\tan x-\frac{{{\tan }^{3}}x}{3}+c\] | |
| 4077. |
\[\int{{{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}\,\,dx}\]= [MP PET 2001] |
| A. | \[\log ({{x}^{4}}+1)+c\] |
| B. | \[\frac{1}{4}\log ({{x}^{4}}+1)+c\] |
| C. | \[-\log ({{x}^{4}}+1)+c\] |
| D. | None of these |
| Answer» C. \[-\log ({{x}^{4}}+1)+c\] | |
| 4078. |
\[\int_{{}}^{{}}{\frac{dx}{2\sqrt{x}(1+x)}=}\] [RPET 2002] |
| A. | \[\frac{1}{2}{{\tan }^{-1}}(\sqrt{x})+c\] |
| B. | \[{{\tan }^{-1}}(\sqrt{x})+c\] |
| C. | \[2{{\tan }^{-1}}(\sqrt{x})+c\] |
| D. | None of these |
| Answer» C. \[2{{\tan }^{-1}}(\sqrt{x})+c\] | |
| 4079. |
The value of \[\int{\frac{2\,\,dx}{\sqrt{1-4{{x}^{2}}}}}\] is [Karnataka CET 2001; Pb. CET 2001] |
| A. | \[{{\tan }^{-1}}(2x)+c\] |
| B. | \[{{\cot }^{-1}}(2x)+c\] |
| C. | \[{{\cos }^{-1}}(2x)+c\] |
| D. | \[{{\sin }^{-1}}(2x)+c\] |
| Answer» E. | |
| 4080. |
\[\int{\frac{1+{{\tan }^{2}}x}{1-{{\tan }^{2}}x}\,dx}\] equals to [RPET 2001] |
| A. | \[\log \left( \frac{1-\tan x}{1+\tan x} \right)+c\] |
| B. | \[\log \left( \frac{1+\tan x}{1-\tan x} \right)+c\] |
| C. | \[\frac{1}{2}\log \left( \frac{1-\tan x}{1+\tan x} \right)+c\] |
| D. | \[\frac{1}{2}\log \left( \frac{1+\tan x}{1-\tan x} \right)+c\] |
| Answer» E. | |
| 4081. |
\[\int_{{}}^{{}}{\frac{{{e}^{m{{\tan }^{-1}}x}}}{1+{{x}^{2}}}dx}\] equals to [RPET 2001] |
| A. | \[{{e}^{{{\tan }^{-1}}x}}\] |
| B. | \[\frac{1}{m}{{e}^{{{\tan }^{-1}}x}}\] |
| C. | \[\frac{1}{m}{{e}^{m{{\tan }^{-1}}x}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 4082. |
\[\int{\frac{dx}{{{({{a}^{2}}+{{x}^{2}})}^{3/2}}}}\] is equal to [RPET 2000] |
| A. | \[\frac{x}{{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] |
| B. | \[\frac{x}{{{a}^{2}}{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] |
| C. | \[\frac{1}{{{a}^{2}}{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{{{a}^{2}}{{\left( {{a}^{2}}+{{x}^{2}} \right)}^{1/2}}}\] | |
| 4083. |
\[\int{{{x}^{x}}(1+\log x)\,\,dx}\] is equal to [RPET 2000] |
| A. | \[{{x}^{x}}\] |
| B. | \[{{x}^{2x}}\] |
| C. | \[{{x}^{x}}\log x\] |
| D. | \[\frac{1}{2}{{(1+\log x)}^{2}}\] |
| Answer» B. \[{{x}^{2x}}\] | |
| 4084. |
\[\int{\frac{{{\sin }^{3}}2x}{{{\cos }^{5}}2x}dx=}\] [Karnataka CET 1999] |
| A. | \[{{\tan }^{4}}x+C\] |
| B. | \[\tan 4x+C\] |
| C. | \[{{\tan }^{4}}2x+x+C\] |
| D. | \[\frac{1}{8}{{\tan }^{4}}2x+C\] |
| Answer» E. | |
| 4085. |
\[\int{\frac{{{e}^{\sqrt{x}}}}{\sqrt{x}}dx}=\] [DCE 1999] |
| A. | \[{{e}^{\sqrt{x}}}\] |
| B. | \[\frac{{{e}^{\sqrt{x}}}}{2}\] |
| C. | \[2\,{{e}^{\sqrt{x}}}\] |
| D. | \[\sqrt{x}\,.\,{{e}^{\sqrt{x}}}\] |
| Answer» D. \[\sqrt{x}\,.\,{{e}^{\sqrt{x}}}\] | |
| 4086. |
What is the value of the integral \[I=\int{\frac{dx}{(1+{{e}^{x}})\,\,(1+{{e}^{-x}})}}\] [DCE 1999] |
| A. | \[\frac{-1}{1+{{e}^{x}}}\] |
| B. | \[\frac{{{e}^{x}}}{1+{{e}^{x}}}\] |
| C. | \[\frac{1}{1+{{e}^{x}}}\] |
| D. | None of these |
| Answer» B. \[\frac{{{e}^{x}}}{1+{{e}^{x}}}\] | |
| 4087. |
The value of \[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{1+{{x}^{4}}}}\ dx}\] is [SCRA 1996] |
| A. | \[{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] |
| B. | \[-{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] |
| C. | \[\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] |
| D. | \[-\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] |
| Answer» D. \[-\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\] | |
| 4088. |
\[\int_{{}}^{{}}{{{e}^{{{x}^{2}}}}x\ dx}\] is equal to [SCRA 1996] |
| A. | \[{{e}^{{{x}^{2}}}}\] |
| B. | \[\frac{1}{2}{{e}^{{{x}^{2}}}}\] |
| C. | \[2{{e}^{{{x}^{2}}}}\] |
| D. | \[\frac{{{e}^{{{x}^{2}}}}-{{x}^{2}}}{2}\] |
| Answer» C. \[2{{e}^{{{x}^{2}}}}\] | |
| 4089. |
\[\int_{{}}^{{}}{{{\sin }^{2}}x\cos x\ dx}\] is equal to [SCRA 1996] |
| A. | \[\frac{{{\cos }^{2}}x}{2}+c\] |
| B. | \[\frac{{{\sin }^{2}}x}{3}+c\] |
| C. | \[\frac{{{\sin }^{3}}x}{3}+c\] |
| D. | \[-\frac{{{\cos }^{2}}x}{2}+c\] |
| Answer» D. \[-\frac{{{\cos }^{2}}x}{2}+c\] | |
| 4090. |
\[\int_{{}}^{{}}{\frac{1}{x}\log x\ dx}\] is equal to [SCRA 1996] |
| A. | \[\frac{1}{2}\log x+c\] |
| B. | \[\frac{1}{2}{{(\log x)}^{2}}+c\] |
| C. | \[\frac{1}{2}\log {{(x)}^{2}}+c\] |
| D. | \[\log x+c\] |
| Answer» C. \[\frac{1}{2}\log {{(x)}^{2}}+c\] | |
| 4091. |
\[\int_{{}}^{{}}{{{\sin }^{3}}x\ dx}\] is equal to [SCRA 1996] |
| A. | \[{{\sin }^{2}}x+1\] |
| B. | \[\sin {{x}^{2}}+{{x}^{2}}+1\] |
| C. | \[\frac{{{\cos }^{3}}x}{3}-\cos x\] |
| D. | \[\frac{1}{4}{{\sin }^{4}}x-\frac{3}{4}{{\sin }^{2}}x\] |
| Answer» D. \[\frac{1}{4}{{\sin }^{4}}x-\frac{3}{4}{{\sin }^{2}}x\] | |
| 4092. |
A primitive of \[\frac{x}{{{x}^{2}}+1}\] is [SCRA 1996] |
| A. | \[{{\log }_{e}}({{x}^{2}}+1)\] |
| B. | \[x{{\tan }^{-1}}x\] |
| C. | \[\frac{{{\log }_{e}}({{x}^{2}}+1)}{2}\] |
| D. | \[\frac{1}{2}x{{\tan }^{-1}}x\] |
| Answer» D. \[\frac{1}{2}x{{\tan }^{-1}}x\] | |
| 4093. |
\[\int_{{}}^{{}}{(x+3){{({{x}^{2}}+6x+10)}^{9}}\ dx}\] equals [SCRA 1996] |
| A. | \[\frac{1}{20}{{({{x}^{2}}+6x+10)}^{10}}+c\] |
| B. | \[\frac{1}{20}{{(x+3)}^{2}}{{({{x}^{2}}+6x+10)}^{10}}+c\] |
| C. | \[\frac{1}{16}{{({{x}^{2}}+6x+10)}^{8}}+c\] |
| D. | \[\frac{1}{38}{{(x+3)}^{19}}+\frac{1}{2}(x+3)+c\] |
| Answer» B. \[\frac{1}{20}{{(x+3)}^{2}}{{({{x}^{2}}+6x+10)}^{10}}+c\] | |
| 4094. |
\[\int_{{}}^{{}}{\frac{{{x}^{e-1}}+{{e}^{x-1}}}{{{x}^{e}}+{{e}^{x}}}dx=}\] |
| A. | \[\log ({{x}^{e}}+{{e}^{x}})+c\] |
| B. | \[e\log ({{x}^{e}}+{{e}^{x}})+c\] |
| C. | \[\frac{1}{e}\log ({{x}^{e}}+{{e}^{x}})+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4095. |
\[\int_{{}}^{{}}{\frac{\cos x-\sin x}{\sqrt{\sin 2x}}\ dx}\] equals [RPET 1996] |
| A. | \[{{\cosh }^{-1}}(\sin x+\cos x)+c\] |
| B. | \[{{\sinh }^{-1}}(\sin x+\cos x)+c\] |
| C. | \[-{{\cosh }^{-1}}(\sin x+\cos x)+c\] |
| D. | \[-{{\sinh }^{-1}}(\sin x+\cos x)+c\] |
| Answer» B. \[{{\sinh }^{-1}}(\sin x+\cos x)+c\] | |
| 4096. |
The value of \[\int_{{}}^{{}}{\left( 1+\frac{1}{{{x}^{2}}} \right)\ {{e}^{\left( x-\frac{1}{x} \right)}}}\ dx\] equals [Kurukshetra CEE 1998] |
| A. | \[{{e}^{x-\frac{1}{x}}}+c\] |
| B. | \[{{e}^{x+\frac{1}{x}}}+c\] |
| C. | \[{{e}^{{{x}^{2}}-\frac{1}{x}}}+c\] |
| D. | \[{{e}^{{{x}^{2}}+\frac{1}{{{x}^{2}}}}}+c\] |
| Answer» B. \[{{e}^{x+\frac{1}{x}}}+c\] | |
| 4097. |
\[\int_{{}}^{{}}{\frac{{{e}^{2x}}+1}{{{e}^{2x}}-1}\ dx}\] equals [RPET 1996] |
| A. | \[\log ({{e}^{x}}-{{e}^{-x}})+c\] |
| B. | \[\log ({{e}^{x}}+{{e}^{-x}})+c\] |
| C. | \[\log ({{e}^{-x}}-{{e}^{x}})+c\] |
| D. | \[\log (1-{{e}^{-x}})+c\] |
| Answer» B. \[\log ({{e}^{x}}+{{e}^{-x}})+c\] | |
| 4098. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}+1}{x({{x}^{2}}-1)}\ dx}\] is equal to [MP PET 1999] |
| A. | \[\log \frac{{{x}^{2}}-1}{x}+c\] |
| B. | \[-\log \frac{{{x}^{2}}-1}{x}+c\] |
| C. | \[\log \frac{x}{{{x}^{2}}+1}+c\] |
| D. | \[-\log \frac{x}{{{x}^{2}}+1}+c\] |
| Answer» B. \[-\log \frac{{{x}^{2}}-1}{x}+c\] | |
| 4099. |
\[\int_{{}}^{{}}{\frac{{{x}^{2}}{{\tan }^{-1}}{{x}^{3}}}{1+{{x}^{6}}}\ dx}\] is equal to [MP PET 1999; UPSEAT 1999] |
| A. | \[{{\tan }^{-1}}({{x}^{3}})+c\] |
| B. | \[\frac{1}{6}{{({{\tan }^{-1}}{{x}^{3}})}^{2}}+c\] |
| C. | \[-\frac{1}{2}{{({{\tan }^{-1}}{{x}^{3}})}^{2}}+c\] |
| D. | \[\frac{1}{2}{{({{\tan }^{-1}}{{x}^{2}})}^{3}}+c\] |
| Answer» C. \[-\frac{1}{2}{{({{\tan }^{-1}}{{x}^{3}})}^{2}}+c\] | |
| 4100. |
If \[\int_{{}}^{{}}{\frac{1}{(1+x)\sqrt{x}}\ dx=f(x)+A}\], where A is any arbitrary constant, then the function \[f(x)\] is [MP PET 1998] |
| A. | \[2{{\tan }^{-1}}x\] |
| B. | \[2{{\tan }^{-1}}\sqrt{x}\] |
| C. | \[2{{\cot }^{-1}}\sqrt{x}\] |
| D. | \[{{\log }_{e}}(1+x)\] |
| Answer» C. \[2{{\cot }^{-1}}\sqrt{x}\] | |