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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.
| 4101. |
\[\int_{{}}^{{}}{x\cos {{x}^{2}}\ dx}\] is equal to [MP PET 1999; Pb. CET 2000] |
| A. | \[-\frac{1}{2}{{\sin }^{2}}x+c\] |
| B. | \[\frac{1}{2}{{\sin }^{2}}x+c\] |
| C. | \[-\frac{1}{2}\sin {{x}^{2}}+c\] |
| D. | \[\frac{1}{2}\sin {{x}^{2}}+c\] |
| Answer» E. | |
| 4102. |
\[\int_{{}}^{{}}{\frac{t}{{{e}^{3{{t}^{2}}}}}\ dt=}\] [MP PET 1997] |
| A. | \[\frac{1}{6}{{e}^{3{{t}^{2}}}}+c\] |
| B. | \[-\frac{1}{6}{{e}^{3{{t}^{2}}}}+c\] |
| C. | \[\frac{1}{6}{{e}^{-3{{t}^{2}}}}+c\] |
| D. | \[-\frac{1}{6}{{e}^{-3{{t}^{2}}}}+c\] |
| Answer» E. | |
| 4103. |
The value of \[\int_{{}}^{{}}{\frac{dx}{x\sqrt{{{x}^{4}}-1}}}\] is |
| A. | \[\frac{1}{2}{{\sec }^{-1}}{{x}^{2}}+k\] |
| B. | \[\log x\sqrt{{{x}^{4}}-1}+k\] |
| C. | \[x\log \sqrt{{{x}^{4}}-1}+k\] |
| D. | \[\log \sqrt{{{x}^{4}}-1}+k\] |
| Answer» B. \[\log x\sqrt{{{x}^{4}}-1}+k\] | |
| 4104. |
The value of\[\int_{{}}^{{}}{{{e}^{x}}{{\sec }^{2}}({{e}^{x}})\ dx}\] is |
| A. | \[\tan ({{e}^{x}})+k\] |
| B. | \[\tan ({{e}^{x}})\ .\ e+k\] |
| C. | \[{{e}^{x}}\tan x+k\] |
| D. | \[\frac{\tan ({{e}^{x}})}{{{e}^{x}}}+k\] |
| Answer» B. \[\tan ({{e}^{x}})\ .\ e+k\] | |
| 4105. |
\[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{{{x}^{2}}+2}}dx=}\] |
| A. | \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}+2{{({{x}^{2}}+2)}^{1/2}}+c\] |
| B. | \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}-2{{({{x}^{2}}+2)}^{1/2}}+c\] |
| C. | \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}+{{({{x}^{2}}+2)}^{1/2}}+c\] |
| D. | \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}-{{({{x}^{2}}+2)}^{1/2}}+c\] |
| Answer» C. \[\frac{1}{3}{{({{x}^{2}}+2)}^{3/2}}+{{({{x}^{2}}+2)}^{1/2}}+c\] | |
| 4106. |
\[\int_{{}}^{{}}{\frac{dx}{1+{{e}^{x}}}=}\] [MP PET 1991; Roorkee 1977] |
| A. | \[\log (1+{{e}^{x}})\] |
| B. | \[-\log (1+{{e}^{-x}})\] |
| C. | \[-\log (1-{{e}^{-x}})\] |
| D. | \[\log ({{e}^{-x}}+{{e}^{-2x}})\] |
| Answer» C. \[-\log (1-{{e}^{-x}})\] | |
| 4107. |
\[\int_{{}}^{{}}{x{{\tan }^{-1}}}xdx=\] [Roorkee 1979] |
| A. | \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x-\frac{1}{2}x+c\] |
| B. | \[\frac{1}{2}({{x}^{2}}-1){{\tan }^{-1}}x-\frac{1}{2}x+c\] |
| C. | \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x+\frac{1}{2}x+c\] |
| D. | \[\frac{1}{2}({{x}^{2}}+1){{\tan }^{-1}}x-x+c\] |
| Answer» B. \[\frac{1}{2}({{x}^{2}}-1){{\tan }^{-1}}x-\frac{1}{2}x+c\] | |
| 4108. |
\[\int_{{}}^{{}}{\left[ \log (\log x)+\frac{1}{{{(\log x)}^{2}}} \right]}\ dx=\] |
| A. | \[x\log (\log x)+\frac{x}{\log x}+c\] |
| B. | \[x\log (\log x)-\frac{x}{\log x}+c\] |
| C. | \[x\log (\log x)+\frac{\log x}{x}+c\] |
| D. | \[x\log (\log x)-\frac{\log x}{x}+c\] |
| Answer» C. \[x\log (\log x)+\frac{\log x}{x}+c\] | |
| 4109. |
If an ant derivative of \[f(x)\] is \[{{e}^{x}}\] and that of \[g(x)\] is \[\cos x,\]then \[\int{f(x)\cos x\,dx}+\int{g(x){{e}^{x}}dx=}\] [Kerala (Engg.) 2005] |
| A. | f(x)g(x)+c |
| B. | f(x)+g(x)+c |
| C. | \[{{e}^{x}}\cos x+c\] |
| D. | f(x) ? g(x)+c |
| E. | \[{{e}^{x}}\cos x+f(x)g(x)+c\] |
| Answer» D. f(x) ? g(x)+c | |
| 4110. |
\[{{I}_{1}}=\int{{{\sin }^{-1}}x\,\,dx}\] and \[{{I}_{2}}=\int{{{\sin }^{-1}}\sqrt{1-{{x}^{2}}}}dx\]then [Kerala (Engg.) 2005] |
| A. | \[{{I}_{1}}={{I}_{2}}\] |
| B. | \[{{I}_{2}}=\pi /2{{I}_{1}}\] |
| C. | \[{{I}_{1}}+{{I}_{2}}=\pi /2x\] |
| D. | \[{{I}_{1}}+{{I}_{2}}=\pi /2\] |
| E. | \[{{I}_{1}}-{{I}_{2}}=\pi /2x\] |
| Answer» D. \[{{I}_{1}}+{{I}_{2}}=\pi /2\] | |
| 4111. |
\[\int_{{}}^{{}}{{{e}^{\sqrt{x}}}\ dx}\] is equal to [MP PET 1998] |
| A. | \[{{e}^{\sqrt{x}}}+A\] |
| B. | \[\frac{1}{2}{{e}^{\sqrt{x}}}+A\] |
| C. | \[2(\sqrt{x}-1){{e}^{\sqrt{x}}}+A\] |
| D. | \[2(\sqrt{x}+1){{e}^{\sqrt{x}}}+A\] (A is an arbitrary constant) |
| Answer» D. \[2(\sqrt{x}+1){{e}^{\sqrt{x}}}+A\] (A is an arbitrary constant) | |
| 4112. |
\[\int_{{}}^{{}}{{{e}^{{{\tan }^{-1}}x}}}\left( \frac{1+x+{{x}^{2}}}{1+{{x}^{2}}} \right)\ dx\] is equal to |
| A. | \[x{{e}^{{{\tan }^{-1}}x}}+c\] |
| B. | \[{{x}^{2}}{{e}^{{{\tan }^{-1}}x}}+c\] |
| C. | \[\frac{1}{x}{{e}^{{{\tan }^{-1}}x}}+c\] |
| D. | None of these |
| Answer» B. \[{{x}^{2}}{{e}^{{{\tan }^{-1}}x}}+c\] | |
| 4113. |
\[\int_{{}}^{{}}{{{\tan }^{-1}}x\,dx=}\] [Roorkee 1977] |
| A. | \[x{{\tan }^{-1}}x+\frac{1}{2}\log (1+{{x}^{2}})\] |
| B. | \[x{{\tan }^{-1}}x-\frac{1}{2}\log (1+{{x}^{2}})\] |
| C. | \[(x-1){{\tan }^{-1}}x\] |
| D. | \[x{{\tan }^{-1}}x-\log (1+{{x}^{2}})\] |
| Answer» C. \[(x-1){{\tan }^{-1}}x\] | |
| 4114. |
\[\int_{{}}^{{}}{{{x}^{5}}.{{e}^{{{x}^{2}}}}dx=}\] |
| A. | \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\] |
| B. | \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}+{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\] |
| C. | \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}-{{e}^{{{x}^{2}}}}+c\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}+{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\] | |
| 4115. |
\[\int_{{}}^{{}}{\frac{x{{\tan }^{-1}}x}{{{(1+{{x}^{2}})}^{3/2}}}\ dx=}\] |
| A. | \[\frac{x+{{\tan }^{-1}}x}{\sqrt{1+{{x}^{2}}}}+c\] |
| B. | \[\frac{x-{{\tan }^{-1}}x}{\sqrt{1+{{x}^{2}}}}+c\] |
| C. | \[\frac{{{\tan }^{-1}}x-x}{\sqrt{1+{{x}^{2}}}}+c\] |
| D. | None of these |
| Answer» C. \[\frac{{{\tan }^{-1}}x-x}{\sqrt{1+{{x}^{2}}}}+c\] | |
| 4116. |
\[\int_{{}}^{{}}{\frac{x{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\ }dx=\] [MNR 1978; EAMCET 1982; IIT 1984] |
| A. | \[x-\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+c\] |
| B. | \[x+\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+c\] |
| C. | \[\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x-x+c\] |
| D. | None of these |
| Answer» B. \[x+\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+c\] | |
| 4117. |
\[\int_{{}}^{{}}{{{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}}dx=}\] [MP PET 1991] |
| A. | \[x{{\tan }^{-1}}x+c\] |
| B. | \[x{{\tan }^{-1}}x-\log (1+{{x}^{2}})+c\] |
| C. | \[2x{{\tan }^{-1}}x+\log (1+{{x}^{2}})+c\] |
| D. | \[2x{{\tan }^{-1}}x-\log (1+{{x}^{2}})+c\] |
| Answer» E. | |
| 4118. |
\[\int_{{}}^{{}}{\cos \sqrt{x}\ dx=}\] [BIT Ranchi 1990; IIT 1977; RPET 1999] |
| A. | \[2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\] |
| B. | \[2[\sqrt{x}\sin \sqrt{x}-\cos \sqrt{x}]+c\] |
| C. | \[2[\cos \sqrt{x}-\sqrt{x}\sin \sqrt{x}]+c\] |
| D. | \[-2[\sqrt{x}\sin \sqrt{x}+\cos \sqrt{x}]+c\] |
| Answer» B. \[2[\sqrt{x}\sin \sqrt{x}-\cos \sqrt{x}]+c\] | |
| 4119. |
\[\int_{{}}^{{}}{{{\sin }^{-1}}(3x-4{{x}^{3}})dx=}\] [AISSE 1986; DSSE 1984] |
| A. | \[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] |
| B. | \[x{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] |
| C. | \[2[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\] |
| D. | \[3[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}]+c\] |
| Answer» E. | |
| 4120. |
\[\int_{{}}^{{}}{32{{x}^{3}}{{(\log x)}^{2}}dx}\] is equal to [MP PET 2004] |
| A. | \[{{x}^{4}}\{8{{(\log x)}^{2}}-4(\log x)+1\}+c\] |
| B. | \[{{x}^{3}}\{{{(\log x)}^{2}}+2\log x\}+c\] |
| C. | \[{{x}^{4}}\{8{{(\log x)}^{2}}-4\log x\}+c\] |
| D. | \[8{{x}^{4}}{{(\log x)}^{2}}+c\] |
| Answer» B. \[{{x}^{3}}\{{{(\log x)}^{2}}+2\log x\}+c\] | |
| 4121. |
\[\int_{{}}^{{}}{\sqrt{x}{{e}^{\sqrt{x}}}\ dx=}\] [Karnataka CET 2004] |
| A. | \[2\sqrt{x}-{{e}^{\sqrt{x}}}-4\sqrt{x}\ {{e}^{\sqrt{x}}}+c\] |
| B. | \[(2x-4\sqrt{x}+4){{e}^{\sqrt{x}}}+c\] |
| C. | \[(2x+4\sqrt{x}+4){{e}^{\sqrt{x}}}+c\] |
| D. | \[(1-4\sqrt{x}){{e}^{\sqrt{x}}}+c\] |
| Answer» C. \[(2x+4\sqrt{x}+4){{e}^{\sqrt{x}}}+c\] | |
| 4122. |
If \[\int_{{}}^{{}}{\frac{{{e}^{x}}(1+\sin x)dx}{1+\cos x}={{e}^{x}}f(x)+c}\], then \[f(x)=\] [RPET 1997; Karnataka CET 2003, 05; Orissa JEE 2004] |
| A. | \[\sin \frac{x}{2}\] |
| B. | \[\cos \frac{x}{2}\] |
| C. | \[\tan \frac{x}{2}\] |
| D. | \[\log \frac{x}{2}\] |
| Answer» D. \[\log \frac{x}{2}\] | |
| 4123. |
\[\int_{{}}^{{}}{x{{\cos }^{2}}}xdx=\] [IIT 1972] |
| A. | \[\frac{{{x}^{4}}}{4}-\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+c\] |
| B. | \[\frac{{{x}^{2}}}{4}+\frac{1}{4}x\sin 2x+\frac{1}{8}\cos 2x+c\] |
| C. | \[\frac{{{x}^{4}}}{4}-\frac{1}{4}x\sin 2x+\frac{1}{8}\cos 2x+c\] |
| D. | \[\frac{{{x}^{4}}}{4}+\frac{1}{4}x\sin 2x-\frac{1}{8}\cos 2x+c\] |
| Answer» C. \[\frac{{{x}^{4}}}{4}-\frac{1}{4}x\sin 2x+\frac{1}{8}\cos 2x+c\] | |
| 4124. |
\[\int_{{}}^{{}}{\frac{x+\sin x}{1+\cos x}\ dx}\] is equal to [Roorkee 1980; UPSEAT 1999] |
| A. | ?\[x\tan \frac{x}{2}+c\] |
| B. | \[x\tan \ \frac{x}{2}+c\] |
| C. | \[x\tan x+c\] |
| D. | \[\frac{1}{2}x\tan x+c\] |
| Answer» C. \[x\tan x+c\] | |
| 4125. |
\[\int_{{}}^{{}}{\frac{x-\sin x}{1-\cos x}dx=}\] [AISSE 1989] |
| A. | \[x\cot \frac{x}{2}+c\] |
| B. | \[-x\cot \frac{x}{2}+c\] |
| C. | \[\cot \frac{x}{2}+c\] |
| D. | None of these |
| Answer» C. \[\cot \frac{x}{2}+c\] | |
| 4126. |
\[\int_{{}}^{{}}{{{\sin }^{-1}}x\ dx}\]is equal to [MP PET 2004] |
| A. | \[\frac{1}{\sqrt{1-{{x}^{2}}}}+c\] |
| B. | \[x{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+c\] |
| C. | \[{{\cos }^{-1}}x+c\] |
| D. | \[x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+c\] |
| Answer» E. | |
| 4127. |
\[\int{{{e}^{x}}(1-\cot x+{{\cot }^{2}}x)\,\,dx}\] equals [MP PET 2002] |
| A. | \[{{e}^{x}}\cot x+c\] |
| B. | \[{{e}^{x}}\text{cosec }x+c\] |
| C. | \[-{{e}^{x}}\cot x+c\] |
| D. | \[-{{e}^{x}}\text{cosec}\,x+c\] |
| Answer» D. \[-{{e}^{x}}\text{cosec}\,x+c\] | |
| 4128. |
\[\int{{{e}^{x}}\left[ f(x)+{f}'(x) \right]\,\,dx}\] is equal to [DCE 2002] |
| A. | \[{{e}^{x}}f(x)\] |
| B. | \[{{e}^{x}}\] |
| C. | \[{{e}^{x}}{f}'(x)\] |
| D. | None of these |
| Answer» B. \[{{e}^{x}}\] | |
| 4129. |
\[\int{\frac{(x+3){{e}^{x}}}{{{(x+4)}^{2}}}\,\,dx=\,\,}\] [Karnataka CET 2000] |
| A. | \[\frac{1}{{{(x+4)}^{2}}}+c\] |
| B. | \[\frac{{{e}^{x}}}{{{(x+4)}^{2}}}+c\] |
| C. | \[\frac{{{e}^{x}}}{x+4}+c\] |
| D. | \[\frac{{{e}^{x}}}{x+3}+c\] |
| Answer» D. \[\frac{{{e}^{x}}}{x+3}+c\] | |
| 4130. |
is equal to [RPET 2000] |
| A. | \[-{{e}^{x}}\tan \,\left( x/2 \right)\] |
| B. | \[-{{e}^{x}}\cot \,\left( x/2 \right)\] |
| C. | \[-\frac{1}{2}{{e}^{x}}\tan \,\left( \frac{x}{2} \right)\] |
| D. | \[\frac{1}{2}{{e}^{x}}\cot \,\left( \frac{x}{2} \right)\] |
| Answer» C. \[-\frac{1}{2}{{e}^{x}}\tan \,\left( \frac{x}{2} \right)\] | |
| 4131. |
\[\int{{{e}^{x}}(1+\tan x+{{\tan }^{2}}x)\,\,dx=}\] [Karnataka CET 1999] |
| A. | \[{{e}^{x}}\sin x+c\] |
| B. | \[{{e}^{x}}\cos x+c\] |
| C. | \[{{e}^{x}}\tan x+c\] |
| D. | \[{{e}^{x}}\sec x+c\] |
| Answer» D. \[{{e}^{x}}\sec x+c\] | |
| 4132. |
\[\int_{{}}^{{}}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)}\,dx=\] [AISSE 1983; MP PET 1994, 96] |
| A. | \[-\frac{{{e}^{x}}}{{{x}^{2}}}+c\] |
| B. | \[\frac{{{e}^{x}}}{{{x}^{2}}}+c\] |
| C. | \[\frac{{{e}^{x}}}{x}+c\] |
| D. | \[-\frac{{{e}^{x}}}{x}+c\] |
| Answer» D. \[-\frac{{{e}^{x}}}{x}+c\] | |
| 4133. |
\[\int_{{}}^{{}}{x\cos x\ dx=}\] [MP PET 1988] |
| A. | \[x\sin x+\cos x\] |
| B. | \[x\sin x-\cos x\] |
| C. | \[x\cos x+\sin x\] |
| D. | \[x\cos x-\sin x\] |
| Answer» B. \[x\sin x-\cos x\] | |
| 4134. |
\[\int_{{}}^{{}}{{{e}^{x}}\frac{({{x}^{2}}+1)}{{{(x+1)}^{2}}}dx=}\] |
| A. | \[\left( \frac{x-1}{x+1} \right){{e}^{x}}+c\] |
| B. | \[{{e}^{x}}\left( \frac{x+1}{x-1} \right)+c\] |
| C. | \[{{e}^{x}}(x+1)(x-1)+c\] |
| D. | None of these |
| Answer» B. \[{{e}^{x}}\left( \frac{x+1}{x-1} \right)+c\] | |
| 4135. |
\[\int_{{}}^{{}}{{{e}^{x}}\left[ {{\sin }^{-1}}\frac{x}{a}+\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right]dx=}\] |
| A. | \[\frac{1}{a}{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] |
| B. | \[a{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] |
| C. | \[{{e}^{x}}{{\sin }^{-1}}\frac{x}{a}+c\] |
| D. | \[\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}+c\] |
| Answer» D. \[\frac{{{e}^{x}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}+c\] | |
| 4136. |
\[\int_{{}}^{{}}{{{e}^{x}}\left[ \frac{1+x\log x}{x} \right]\ dx=}\] |
| A. | \[{{e}^{x}}+\log x+c\] |
| B. | \[\frac{{{e}^{x}}}{\log x}+c\] |
| C. | \[{{e}^{x}}-\log x+c\] |
| D. | \[{{e}^{x}}\log x+c\] |
| Answer» E. | |
| 4137. |
\[\int_{{}}^{{}}{\frac{{{e}^{x}}(x-1)}{{{x}^{2}}}\ dx=}\] |
| A. | \[\frac{1}{x}{{e}^{x}}+c\] |
| B. | \[x{{e}^{-x}}+c\] |
| C. | \[\frac{1}{{{x}^{2}}}{{e}^{x}}+c\] |
| D. | \[\left( x-\frac{1}{x} \right){{e}^{x}}+c\] |
| Answer» B. \[x{{e}^{-x}}+c\] | |
| 4138. |
\[\int_{{}}^{{}}{{{e}^{2x}}\frac{1+\sin 2x}{1+\cos 2x}}\ dx=\] |
| A. | \[{{e}^{2x}}\tan x+c\] |
| B. | \[{{e}^{2x}}\cot x+c\] |
| C. | \[\frac{{{e}^{2x}}\tan x}{2}+c\] |
| D. | \[\frac{{{e}^{2x}}\cot x}{2}+c\] |
| Answer» D. \[\frac{{{e}^{2x}}\cot x}{2}+c\] | |
| 4139. |
\[\int_{{}}^{{}}{{{e}^{x}}\sin x(\sin x+2\cos x)}\ dx=\] [MP PET 1988] |
| A. | \[{{e}^{x}}{{\sin }^{2}}x+c\] |
| B. | \[{{e}^{x}}\sin x+c\] |
| C. | \[{{e}^{x}}\sin 2x+c\] |
| D. | None of these |
| Answer» B. \[{{e}^{x}}\sin x+c\] | |
| 4140. |
\[\int_{{}}^{{}}{{{e}^{x}}[\tan x-\log (\cos x)]\ dx=}\] [MP PET 1991] |
| A. | \[{{e}^{x}}\log (\sec x)+c\] |
| B. | \[{{e}^{x}}\log (\cos \text{ec}x)+c\] |
| C. | \[{{e}^{x}}\log (\cos x)+c\] |
| D. | \[{{e}^{x}}\log (\sin x)+c\] |
| Answer» B. \[{{e}^{x}}\log (\cos \text{ec}x)+c\] | |
| 4141. |
\[\int_{{}}^{{}}{\frac{x{{e}^{x}}}{{{(1+x)}^{2}}}dx=}\] [MP PET 1997; UPSEAT 2001; RPET 2002] |
| A. | \[\frac{{{e}^{-x}}}{1+x}+c\] |
| B. | \[-\frac{{{e}^{-x}}}{1+x}+c\] |
| C. | \[\frac{{{e}^{x}}}{1+x}+c\] |
| D. | \[-\frac{{{e}^{x}}}{1+x}+c\] |
| Answer» D. \[-\frac{{{e}^{x}}}{1+x}+c\] | |
| 4142. |
\[\int_{{}}^{{}}{{{e}^{x}}(1+\tan x)\sec x\ dx=}\] [Karnataka CET 2005] |
| A. | \[{{e}^{x}}\cot x\] |
| B. | \[{{e}^{x}}\tan x\] |
| C. | \[{{e}^{x}}\sec x\] |
| D. | \[{{e}^{x}}\cos x\] |
| Answer» D. \[{{e}^{x}}\cos x\] | |
| 4143. |
\[\int{\cos ({{\log }_{e}}x)\,dx}\] is equal to [MP PET 2003] |
| A. | \[\frac{1}{2}x\{\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)\}\] |
| B. | \[x\{\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)\}\] |
| C. | \[\frac{1}{2}x\{\cos ({{\log }_{e}}x)-\sin ({{\log }_{e}}x)\}\] |
| D. | \[x\{\cos ({{\log }_{e}}x)-\sin ({{\log }_{e}}x)\}\] |
| Answer» B. \[x\{\cos ({{\log }_{e}}x)+\sin ({{\log }_{e}}x)\}\] | |
| 4144. |
\[\int_{{}}^{{}}{x\log xdx=}\] [MP PET 1987] |
| A. | \[\frac{{{x}^{2}}}{2}\log x-\frac{{{x}^{2}}}{2}+c\] |
| B. | \[\frac{{{x}^{2}}}{2}\log x-\frac{{{x}^{2}}}{4}+c\] |
| C. | \[\frac{{{x}^{2}}}{2}\log x+\frac{{{x}^{2}}}{2}+c\] |
| D. | None of these |
| Answer» C. \[\frac{{{x}^{2}}}{2}\log x+\frac{{{x}^{2}}}{2}+c\] | |
| 4145. |
\[\int{{{\cos }^{-1}}\left( \frac{1}{x} \right)\,\,dx}\] [RPET 2002] |
| A. | \[x{{\sec }^{-1}}x+{{\cosh }^{-1}}x+C\] |
| B. | \[x{{\sec }^{-1}}x-{{\cosh }^{-1}}x+C\] |
| C. | \[x{{\sec }^{-1}}x-{{\sin }^{-1}}x+C\] |
| D. | None of these |
| Answer» C. \[x{{\sec }^{-1}}x-{{\sin }^{-1}}x+C\] | |
| 4146. |
\[\int{{{x}^{3}}\log x\,\,dx=}\] [Karnataka CET 2002] |
| A. | \[\frac{{{x}^{4}}\log x}{4}+c\] |
| B. | \[\frac{1}{16}[4{{x}^{4}}\log x-{{x}^{4}}]+c\] |
| C. | \[\frac{1}{8}[{{x}^{4}}\log x-4{{x}^{2}}]+c\] |
| D. | \[\frac{1}{16}[4{{x}^{4}}\log x+{{x}^{4}}]+c\] |
| Answer» C. \[\frac{1}{8}[{{x}^{4}}\log x-4{{x}^{2}}]+c\] | |
| 4147. |
If \[\frac{d}{dx}f(x)=x\cos x+\sin x\] and \[f(0)=2\], then \[f(x)=\] [MP PET 1989] |
| A. | \[x\sin x\] |
| B. | \[x\cos x+\sin x+2\] |
| C. | \[x\sin x+2\] |
| D. | \[x\cos x+2\] |
| Answer» D. \[x\cos x+2\] | |
| 4148. |
If \[\int{x{{e}^{2x}}\,\,dx}\] is equal to \[{{e}^{2x}}f(x)+C\] where C is constant of integration, then f(x) is [UPSEAT 2001] |
| A. | \[(3x-1)/4\] |
| B. | \[(2x+1)/2\] |
| C. | \[(2x-1)/4\] |
| D. | \[(x-4)/6\] |
| Answer» D. \[(x-4)/6\] | |
| 4149. |
The value of \[\int{\frac{\log x}{{{(x+1)}^{2}}}dx}\] is [UPSEAT 1999] |
| A. | \[\frac{-\log x}{x+1}+\log x-\log \,(x+1)\] |
| B. | \[\frac{\log x}{\left( x+1 \right)}+\log x-\log \,(x+1)\] |
| C. | \[\frac{\log x}{x+1}-\log x-\log \,(x+1)\] |
| D. | \[\frac{-\log x}{x+1}-\log x-\log \,(x+1)\] |
| Answer» B. \[\frac{\log x}{\left( x+1 \right)}+\log x-\log \,(x+1)\] | |
| 4150. |
\[\int_{{}}^{{}}{{{x}^{3}}{{e}^{{{x}^{2}}}}dx=}\] [MNR 1980] |
| A. | \[\frac{1}{2}({{x}^{2}}+1){{e}^{{{x}^{2}}}}+c\] |
| B. | \[({{x}^{2}}+1){{e}^{{{x}^{2}}}}+c\] |
| C. | \[\frac{1}{2}({{x}^{2}}-1){{e}^{{{x}^{2}}}}+c\] |
| D. | \[({{x}^{2}}-1){{e}^{{{x}^{2}}}}+c\] |
| Answer» D. \[({{x}^{2}}-1){{e}^{{{x}^{2}}}}+c\] | |