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.
| 8051. |
The equation \[{{\tan }^{-1}}(1+x)+ta{{n}^{-1}}(1-x)=\frac{\pi }{2}\] is satisfied by |
| A. | \[x=1\] |
| B. | \[x=-1\] |
| C. | \[x=0\] |
| D. | \[x=\frac{1}{2}\] |
| Answer» D. \[x=\frac{1}{2}\] | |
| 8052. |
If \[{{\sin }^{-1}}(x-1)+co{{s}^{-1}}(x-3)+ta{{n}^{-1}}\left( \frac{x}{2-{{x}^{2}}} \right)\]\[={{\cos }^{-1}}k+\pi ,\] Then the value of k is |
| A. | 1 |
| B. | \[-\frac{1}{\sqrt{2}}\] |
| C. | \[\frac{1}{\sqrt{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8053. |
The domain of the function\[f(x)=si{{n}^{-1}}\left\{ {{\log }_{2}}\left( \frac{1}{2}{{x}^{2}} \right) \right\}\] is |
| A. | \[[-2,-1)\cup [1,2]\] |
| B. | \[(-2,-1]\cup [1,2]\] |
| C. | \[[-2,-1]\cup [1,2]\] |
| D. | \[(-2,-1)\cup (1,2)\] |
| Answer» D. \[(-2,-1)\cup (1,2)\] | |
| 8054. |
The value of \[\sin {{\cot }^{-1}}\tan {{\cos }^{-1}}x,\] is |
| A. | \[x\] |
| B. | \[\frac{1}{x}\] |
| C. | \[1\] |
| D. | \[0\] |
| Answer» B. \[\frac{1}{x}\] | |
| 8055. |
\[f(x)=ta{{n}^{-1}}x+{{\tan }^{-1}}\left( \frac{1}{x} \right);g(x)=si{{n}^{-1}}x+co{{x}^{-1}}x\]are identical functions if |
| A. | \[x\in R\] |
| B. | \[x>0\] |
| C. | \[x\in [-1,1]\] |
| D. | \[x\in (0,1]\] |
| Answer» E. | |
| 8056. |
There exists a positive real number x satisfying \[\cos (ta{{n}^{-1}}x)=x\], Then the value of \[{{\cos }^{-1}}\left( \frac{{{x}^{2}}}{2} \right)\]is |
| A. | \[\frac{\pi }{10}\] |
| B. | \[\frac{\pi }{5}\] |
| C. | \[\frac{2\pi }{5}\] |
| D. | \[\frac{4\pi }{5}\] |
| Answer» D. \[\frac{4\pi }{5}\] | |
| 8057. |
If \[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha \] then \[4{{x}^{2}}-2xy\,\,\cos \alpha +{{y}^{2}}\] is equal to |
| A. | 2 sin \[\alpha \] |
| B. | 4 |
| C. | \[4{{\sin }^{2}}\alpha \] |
| D. | \[-\,4{{\sin }^{2}}\alpha \] |
| Answer» D. \[-\,4{{\sin }^{2}}\alpha \] | |
| 8058. |
If \[{{\tan }^{-1}}(si{{n}^{2}}\theta -2sin\theta +3)+co{{t}^{-1}}({{5}^{{{\sec }^{2}}y}}+1)=\frac{\pi }{2}\], then the value of \[{{\cos }^{2}}\theta -\sin \theta \]is equal to |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | none of these |
| Answer» D. none of these | |
| 8059. |
\[\tan \left[ \frac{1}{2}{{\sin }^{-1}}\left( \frac{2a}{1+{{a}^{2}}} \right)+\frac{1}{2}{{\cos }^{-1}}\left( \frac{1-{{a}^{2}}}{1+{{a}^{2}}} \right) \right]=\] |
| A. | \[\frac{2a}{1+{{a}^{2}}}\] |
| B. | \[\frac{1-{{a}^{2}}}{1+{{a}^{2}}}\] |
| C. | \[\frac{2a}{1-{{a}^{2}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8060. |
If \[\sin \left( {{\sin }^{-1}}\frac{1}{5}+{{\cos }^{-1}}x \right)=1\],then x is equal to [MNR 1994; Kerala (Engg.) 2005] |
| A. | 1 |
| B. | 0 |
| C. | \[\frac{4}{5}\] |
| D. | \[\frac{1}{5}\] |
| Answer» E. | |
| 8061. |
If \[{{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}A,\]then A = [MP PET 1988] |
| A. | \[x-y\] |
| B. | \[x+y\] |
| C. | \[\frac{x-y}{1+xy}\] |
| D. | \[\frac{x+y}{1-xy}\] |
| Answer» D. \[\frac{x+y}{1-xy}\] | |
| 8062. |
What is the value of\[\int_{1}^{2}{{{e}^{x}}\left( \frac{1}{x}-\frac{1}{{{x}^{2}}} \right)dx}\]? |
| A. | \[e\left( \frac{e}{2}-1 \right)\] |
| B. | \[e(e-1)\] |
| C. | \[e-\frac{1}{e}\] |
| D. | 0 |
| Answer» B. \[e(e-1)\] | |
| 8063. |
If \[f(x)\] is an even function, then what is \[\int\limits_{0}^{\pi }{f(\cos x)dx}\] equal to? |
| A. | 0 |
| B. | \[\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\] |
| C. | \[2\int\limits_{0}^{\frac{\pi }{2}}{f(\cos x)dx}\] |
| D. | 1 |
| Answer» D. 1 | |
| 8064. |
If\[\int{\frac{dx}{x({{x}^{n}}+1)}=A\,\,\log \left| \frac{{{x}^{n}}+1}{{{x}^{n}}} \right|+B,B\in R}\]. Then |
| A. | \[A=\frac{1}{2}\] |
| B. | \[A=-1\] |
| C. | \[A=-\frac{1}{n}\] |
| D. | \[A=\frac{1}{2n}\] |
| Answer» D. \[A=\frac{1}{2n}\] | |
| 8065. |
The value of \[\int_{0}^{\pi }{ln(1+\cos \,\,x)dx}\] is |
| A. | \[\frac{\pi }{2}\log 2\] |
| B. | \[\pi \log 2\] |
| C. | \[-\pi \log 2\] |
| D. | 0 |
| Answer» D. 0 | |
| 8066. |
If\[\int{f(x)dx=g(x)+c}\], then \[\int{{{f}^{-1}}(x)dx}\] is equal to |
| A. | \[x{{f}^{-1}}(x)+C\] |
| B. | \[f({{g}^{-1}})(x))+C\] |
| C. | \[x{{f}^{-1}}(x)-g({{f}^{-1}})(x))+C\] |
| D. | \[{{g}^{-1}}(x)+C\] |
| Answer» D. \[{{g}^{-1}}(x)+C\] | |
| 8067. |
\[\int\limits_{-\pi /2}^{\pi /2}{\frac{ln\,(\cos x)}{1+{{e}^{x}}.{{e}^{\sin \,\,x}}}dx=}\] |
| A. | \[-2\pi \,\,ln\,\,2\] |
| B. | \[-\frac{\pi }{4}ln\,\,2\] |
| C. | \[-\pi \,\,ln\,\,2\] |
| D. | \[-\frac{\pi }{2}ln\,\,2\] |
| Answer» E. | |
| 8068. |
If \[{{u}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}\theta }d\theta \] then \[{{u}_{n}}+{{u}_{n-2}}\] is: |
| A. | \[\frac{1}{n-1}\] |
| B. | \[\frac{1}{n+1}\] |
| C. | \[\frac{1}{2n-1}\] |
| D. | \[\frac{1}{2n+1}\] |
| Answer» B. \[\frac{1}{n+1}\] | |
| 8069. |
If \[{{I}_{1}}=\int\limits_{0}^{\frac{\pi }{2}}{\cos (\sin \,\,x)dx;{{I}_{2}}=\int\limits_{0}^{\frac{\pi }{2}}{\sin (\cos \,\,x)dx}}\] and \[{{I}_{3}}=\int\limits_{0}^{\frac{\pi }{2}}{\cos x\,\,dx,}\] then |
| A. | \[{{I}_{1}}>{{I}_{3}}>{{I}_{2}}\] |
| B. | \[{{I}_{3}}>{{I}_{1}}>{{I}_{2}}\] |
| C. | \[{{I}_{1}}>{{I}_{2}}>{{I}_{3}}\] |
| D. | \[{{I}_{3}}>{{I}_{2}}>{{I}_{1}}\] |
| Answer» B. \[{{I}_{3}}>{{I}_{1}}>{{I}_{2}}\] | |
| 8070. |
\[\int{\frac{(1+x)}{x{{(1+x{{e}^{x}})}^{2}}}dx}\] is |
| A. | \[\ln \,\,\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+\frac{1}{1+x{{e}^{x}}}+C\] |
| B. | \[(1+x{{e}^{x}})+ln\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+C\] |
| C. | \[\frac{1}{1+x{{e}^{x}}}+ln\left| x{{e}^{x}}(1+x{{e}^{x}}) \right|+C\] |
| D. | None of these |
| Answer» B. \[(1+x{{e}^{x}})+ln\left| \frac{x{{e}^{x}}}{1+x{{e}^{x}}} \right|+C\] | |
| 8071. |
If \[\int{{{\sin }^{3}}x{{\cos }^{5}}xdx}\]\[=A{{\sin }^{4}}x+B{{\sin }^{6}}x+C{{\sin }^{8}}x+D\] Then |
| A. | \[A=\frac{1}{4},B=-\frac{1}{3},C=\frac{1}{8},D\in R\] |
| B. | \[A=\frac{1}{8},B=\frac{1}{4},C=\frac{1}{3},D\in R\] |
| C. | \[A=0,B=-\frac{1}{6},C=\frac{1}{8},D\in R\] |
| D. | None of these |
| Answer» B. \[A=\frac{1}{8},B=\frac{1}{4},C=\frac{1}{3},D\in R\] | |
| 8072. |
Evaluate: \[\int{\frac{1}{1+3{{\sin }^{2}}x+8{{\cos }^{2}}x}dx}\] |
| A. | \[\frac{1}{6}{{\tan }^{-1}}(2\tan x)+C\] |
| B. | \[{{\tan }^{-1}}(2\tan x)+C\] |
| C. | \[\frac{1}{6}{{\tan }^{-1}}\left( \frac{2\tan x}{3} \right)+C\] |
| D. | None of these |
| Answer» D. None of these | |
| 8073. |
What is \[\int{{{\sec }^{n}}x\tan xdx}\] equal to? |
| A. | \[\frac{{{\sec }^{n}}x}{n}+c\] |
| B. | \[\frac{{{\sec }^{n-1}}x}{n-1}+c\] |
| C. | \[\frac{{{\tan }^{n}}x}{n}+c\] |
| D. | \[\frac{{{\tan }^{n-1}}x}{n-1}+c\] Where ?c? is a constant of integration. |
| Answer» B. \[\frac{{{\sec }^{n-1}}x}{n-1}+c\] | |
| 8074. |
If \[\int{g(x)dx=g(x),}\] then\[\int{g(x)\{f(x)+f'(x)\}dx}\] is equal to |
| A. | \[g(x)f(x)-g(x)f'(x)+C\] |
| B. | \[g(x)f'(x)+C\] |
| C. | \[g(x)f(x)+C\] |
| D. | \[g(x){{f}^{2}}(x)+C\] |
| Answer» D. \[g(x){{f}^{2}}(x)+C\] | |
| 8075. |
If \[\int{x\log \left( 1+\frac{1}{x} \right)dx}\]\[=f(x)\log (x+1)+g(x){{x}^{2}}+Lx+C\], then |
| A. | \[f(x)=\frac{1}{2}{{x}^{2}}\] |
| B. | \[g(x)=\log x\] |
| C. | \[L=1\] |
| D. | None of these |
| Answer» E. | |
| 8076. |
\[{{A}_{n}}=\int\limits_{0}^{\pi /2}{\frac{\sin (2n-1)x}{\sin x}}dx;{{B}_{n}}=\int\limits_{0}^{\pi /2}{{{\left( \frac{\sin nx}{\sin x} \right)}^{2}}dx;}\] For \[n\in N,\] then |
| A. | \[{{A}_{n+1}}={{A}_{n}},{{B}_{n+1}}-{{B}_{n}}={{A}_{n+1}}\] |
| B. | \[{{B}_{n+1}}={{B}_{n}}\] |
| C. | \[{{A}_{n+1}}-{{A}_{n}}={{B}_{n+1}}\] |
| D. | None of these |
| Answer» B. \[{{B}_{n+1}}={{B}_{n}}\] | |
| 8077. |
If \[{{l}^{r}}(x)\] means log log log ??.x, the log being repeated r times. then \[\int{\{xl(x){{l}^{2}}(x){{l}^{3}}(x)....{{l}^{r}}(x)\}{{-}^{1}}dx}\] is equal to |
| A. | \[{{l}^{r+1}}(x)+C\] |
| B. | \[\frac{{{l}^{r+1}}(x)}{r+1}+C\] |
| C. | \[{{l}^{r}}(x)+C\] |
| D. | None |
| Answer» B. \[\frac{{{l}^{r+1}}(x)}{r+1}+C\] | |
| 8078. |
If \[{{I}_{1}}=\int\limits_{0}^{\pi }{xf({{\sin }^{3}}x+{{\cos }^{2}}x)dx}\] and \[{{I}_{2}}=\pi \int\limits_{0}^{\pi /2}{f({{\sin }^{3}}x+{{\cos }^{2}}x)dx}\], then |
| A. | \[{{I}_{1}}=2{{I}_{2}}\] |
| B. | \[2{{I}_{1}}={{I}_{2}}\] |
| C. | \[{{I}_{1}}={{I}_{2}}\] |
| D. | \[{{I}_{1}}+{{I}_{2}}=0\] |
| Answer» D. \[{{I}_{1}}+{{I}_{2}}=0\] | |
| 8079. |
\[\underset{n\to \infty }{\mathop{Lim}}\,{{\left\{ \frac{n!}{{{(kn)}^{n}}} \right\}}^{\frac{1}{n}}},\] where \[k\ne 0\] is a constant and \[n\in N\] is equal to |
| A. | ke |
| B. | \[{{k}^{-1}}e\] |
| C. | \[k{{e}^{-1}}\] |
| D. | \[{{k}^{-1}}{{e}^{-1}}\] |
| Answer» E. | |
| 8080. |
If\[f(x)=ln(x-\sqrt{1+{{x}^{2}}})\], then what is \[\int{f''(x)dx}\] equal to? |
| A. | \[\frac{1}{(x-\sqrt{1+{{x}^{2}}})}+c\] |
| B. | \[-\frac{1}{\sqrt{1+{{x}^{2}}}}+c\] |
| C. | \[-\sqrt{1+{{x}^{2}}}+c\] |
| D. | \[\text{ln}\,(x-\sqrt{1+{{x}^{2}}})+c\] |
| Answer» C. \[-\sqrt{1+{{x}^{2}}}+c\] | |
| 8081. |
\[\int{\sqrt{\frac{x}{1-x}}dx}\] is equal to |
| A. | \[{{\sin }^{-1}}\sqrt{x}+c\] |
| B. | \[{{\sin }^{-1}}\{\sqrt{x}-\sqrt{x(1-x)\}}+c\] |
| C. | \[{{\sin }^{-1}}\sqrt{x(1-x)}+c\] |
| D. | \[{{\sin }^{-1}}\sqrt{x}-\sqrt{x(1-x)}+c\] |
| Answer» E. | |
| 8082. |
What is \[\int{\frac{{{e}^{x}}(1+x)}{{{\cos }^{2}}\left( x{{e}^{x}} \right)}dx}\] equal to? |
| A. | \[x{{e}^{x}}+c\] |
| B. | \[\cos (x{{e}^{x}})+c\] |
| C. | \[\tan (x{{e}^{x}})+c\] |
| D. | \[x\cos ec(x{{e}^{x}})+c\] Where c is a constant of integration. |
| Answer» D. \[x\cos ec(x{{e}^{x}})+c\] Where c is a constant of integration. | |
| 8083. |
\[\int{\frac{{{x}^{2}}}{({{x}^{2}}+1)({{x}^{2}}+4)}dx}\] is equal to |
| A. | \[{{\tan }^{-1}}x+2{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\] |
| B. | \[{{\tan }^{-1}}\left( \frac{x}{2} \right)-4{{\tan }^{-1}}x+C\] |
| C. | \[-\frac{1}{3}{{\tan }^{-1}}x+\frac{2}{3}{{\tan }^{-1}}\left( \frac{x}{2} \right)+C\] |
| D. | \[4{{\tan }^{-1}}\left( \frac{x}{2} \right)-2{{\tan }^{-1}}x+C\] |
| Answer» D. \[4{{\tan }^{-1}}\left( \frac{x}{2} \right)-2{{\tan }^{-1}}x+C\] | |
| 8084. |
If \[\int{f(x)\sin x\cos x\,\,dx=\frac{1}{2({{b}^{2}}-{{a}^{2}})}{{\log }_{e}}(f(x))+A,}\]\[b\ne \pm a,\] then \[{{\{f(x)\}}^{-1}}\] is equal to |
| A. | \[{{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x+C\] |
| B. | \[{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x+C\] |
| C. | \[{{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}si{{n}^{2}}x+C\] |
| D. | \[{{a}^{2}}{{\cos }^{2}}x-{{b}^{2}}si{{n}^{2}}x+C\] |
| Answer» B. \[{{a}^{2}}{{\sin }^{2}}x-{{b}^{2}}{{\cos }^{2}}x+C\] | |
| 8085. |
Find the value of \[\int\limits_{0}^{9}{[\sqrt{x}+2]dx}\] where \[[\cdot ]\] is the greatest integer function: |
| A. | 31 |
| B. | 22 |
| C. | 23 |
| D. | None of these |
| Answer» B. 22 | |
| 8086. |
If \[\int {{x}^{5}}{{(1+{{x}^{3}})}^{2/3}}dx=A{{(1+{{x}^{3}})}^{8/3}}+B{{(1+{{x}^{3}})}^{5/3}}+c\],then |
| A. | \[A=\frac{1}{4},B=\frac{1}{5}\] |
| B. | \[A=\frac{1}{8},B=-\frac{1}{5}\] |
| C. | \[A=-\frac{1}{8},B=\frac{1}{5}\] |
| D. | none of these |
| Answer» C. \[A=-\frac{1}{8},B=\frac{1}{5}\] | |
| 8087. |
The value of \[\int\limits_{1}^{e}{\frac{1+{{x}^{2}}\ln \,x}{x+{{x}^{2}}\ln \,x}dx}\]is |
| A. | \[e\] |
| B. | \[\ln \,(1+e)\] |
| C. | \[e+\ln (1+e)\] |
| D. | \[e-\ln (1+e)\] |
| Answer» E. | |
| 8088. |
\[\int_{{}}^{{}}{(x+3){{({{x}^{2}}+6x+10)}^{9}}\ dx}\] 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. \[\frac{1}{20}{{(x+3)}^{2}}{{({{x}^{2}}+6x+10)}^{10}}+c\] | |
| 8089. |
\[\int_{{}}^{{}}{\frac{\cos \sqrt{x}}{\sqrt{x}}}dx=\] [MP PET 1987; IIT 1990; SCRA 1996; RPET 2001] |
| A. | \[2\cos \sqrt{x}+c\] |
| B. | \[2\sin \sqrt{x}+c\] |
| C. | \[\sin \sqrt{x}+c\] |
| D. | \[\frac{1}{2}\cos \sqrt{x}+c\] |
| Answer» C. \[\sin \sqrt{x}+c\] | |
| 8090. |
\[\int_{{}}^{{}}{\frac{1}{{{\cos }^{2}}x{{(1-\tan x)}^{2}}}dx=}\] |
| A. | \[\frac{1}{\tan x-1}+c\] |
| B. | \[\frac{1}{1-\tan x}+c\] |
| C. | \[-\frac{1}{3}\frac{1}{{{(1-\tan x)}^{3}}}+c\] |
| D. | None of these |
| Answer» C. \[-\frac{1}{3}\frac{1}{{{(1-\tan x)}^{3}}}+c\] | |
| 8091. |
\[\int_{{}}^{{}}{\frac{\sin x\ dx}{{{a}^{2}}+{{b}^{2}}{{\cos }^{2}}x}}=\] |
| A. | \[\log ({{a}^{2}}+{{b}^{2}}{{\cos }^{2}}x)+c\] |
| B. | \[\frac{1}{ab}{{\tan }^{-1}}\left( \frac{a\cos x}{b} \right)+c\] |
| C. | \[\frac{1}{ab}{{\cot }^{-1}}\left( \frac{b\cos x}{a} \right)+c\] |
| D. | \[\frac{1}{ab}{{\cot }^{-1}}\left( \frac{a\cos x}{b} \right)+c\] |
| Answer» D. \[\frac{1}{ab}{{\cot }^{-1}}\left( \frac{a\cos x}{b} \right)+c\] | |
| 8092. |
\[\int{\frac{1+{{\tan }^{2}}x}{1-{{\tan }^{2}}x}\,dx}\] 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. | |
| 8093. |
\[\int_{{}}^{{}}{\frac{dx}{x[{{(\log x)}^{2}}+4\log x-1]}}=\] |
| A. | \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\] |
| B. | \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\] |
| C. | \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\] |
| D. | \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\] |
| Answer» B. \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\] | |
| 8094. |
\[\int_{{}}^{{}}{\sqrt{{{x}^{2}}-8x+7}}\ dx=\] |
| A. | \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}+9\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\] |
| B. | \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}-3\sqrt{2}\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\] |
| C. | \[\frac{1}{2}(x-4)\sqrt{{{x}^{2}}-8x+7}-\frac{9}{2}\log [x-4+\sqrt{{{x}^{2}}-8x+7}]+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 8095. |
\[\int{\sqrt{\frac{1+x}{1-x}}\,\,dx=}\] [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. \[-{{\sin }^{-1}}x-\sqrt{{{x}^{2}}-1}\,+c\] | |
| 8096. |
\[\int_{{}}^{{}}{\frac{1}{{{[{{(x-1)}^{3}}{{(x+2)}^{5}}]}^{1/4}}}\ dx}\] is equal to |
| A. | \[\frac{4}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\] |
| B. | \[\frac{4}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\] |
| C. | \[\frac{1}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\] |
| D. | \[\frac{1}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\] |
| Answer» B. \[\frac{4}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\] | |
| 8097. |
\[\int_{{}}^{{}}{{{e}^{-2x}}\sin 3x\ dx=}\] |
| A. | \[\frac{1}{13}{{e}^{-2x}}[\sin 3x+\cos 3x]+c\] |
| B. | \[-\frac{1}{13}{{e}^{-2x}}[\sin 3x+\cos 3x]+c\] |
| C. | \[\frac{1}{13}{{e}^{-2x}}[2\sin 3x+3\cos 3x]+c\] |
| D. | \[-\frac{1}{13}{{e}^{-2x}}[2\sin 3x+3\cos 3x]+c\] |
| Answer» E. | |
| 8098. |
\[\int_{{}}^{{}}{\left[ \frac{1}{\log x}-\frac{1}{{{(\log x)}^{2}}} \right]dx=}\] |
| A. | \[\frac{1}{\log x}+c\] |
| B. | \[\frac{x}{\log x}+c\] |
| C. | \[\frac{x}{{{(\log x)}^{2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{x}{{{(\log x)}^{2}}}\] | |
| 8099. |
\[\int{{{\cos }^{-1}}\left( \frac{1}{x} \right)\,\,dx}\] [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. \[x{{\sec }^{-1}}x-{{\sin }^{-1}}x+C\] | |
| 8100. |
If \[\int{\sqrt{2}\sqrt{1+\sin x}}\,\,dx=-\,4\cos (ax+b)+c\] then the value of (a, b) is [UPSEAT 2002] |
| A. | \[\frac{1}{2},\,\frac{\pi }{4}\] |
| B. | \[1,\,\frac{\pi }{2}\] |
| C. | 1, 1 |
| D. | None of these |
| Answer» B. \[1,\,\frac{\pi }{2}\] | |