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.
| 1051. |
\[\int\limits_{0}^{2\pi }{\log \left( \frac{a+b\sec x}{a-b\sec x} \right)}dx=\] |
| A. | 0 |
| B. | \[\pi /2\] |
| C. | \[\frac{\pi (a+b)}{a-b}\] |
| D. | \[\frac{\pi }{2}({{a}^{2}}-{{b}^{2}})\] |
| Answer» B. \[\pi /2\] | |
| 1052. |
\[I=\int{\left\{ {{\log }_{e}}{{\log }_{e}}x+\frac{1}{{{({{\log }_{e}}x)}^{2}}} \right\}dx}\] is equal to: |
| A. | \[x{{\log }_{e}}{{\log }_{e}}x+c\] |
| B. | \[x{{\log }_{e}}{{\log }_{e}}x-\frac{x}{{{\log }_{e}}x}+c\] |
| C. | \[x{{\log }_{e}}{{\log }_{e}}x+\frac{x}{{{\log }_{e}}x}+c\] |
| D. | None of these. |
| Answer» C. \[x{{\log }_{e}}{{\log }_{e}}x+\frac{x}{{{\log }_{e}}x}+c\] | |
| 1053. |
\[\int\limits_{0}^{\infty }{\left[ \frac{2}{{{e}^{x}}} \right]}\,dx\] is equal to ([x] = greatest integer \[\le \] x) |
| A. | \[{{\log }_{e}}2\] |
| B. | \[{{e}^{2}}\] |
| C. | 0 |
| D. | \[\frac{2}{e}\] |
| Answer» B. \[{{e}^{2}}\] | |
| 1054. |
If \[{{I}_{n}}=\int\limits_{0}^{\frac{\pi }{4}}{{{\tan }^{n}}x\,dx}\] then what is \[{{I}_{n}}+{{I}_{n-2}}\] equal to? |
| A. | \[\frac{1}{n}\] |
| B. | \[\frac{1}{(n-1)}\] |
| C. | \[\frac{n}{(n-1)}\] |
| D. | \[\frac{1}{(n-2)}\] |
| Answer» C. \[\frac{n}{(n-1)}\] | |
| 1055. |
What is the value of\[\int_{0}^{1}{x{{e}^{{{x}^{2}}}}dx}\]? |
| A. | \[\frac{(e-1)}{2}\] |
| B. | \[{{e}^{2}}-1\] |
| C. | \[2(e-1)\] |
| D. | \[e-1\] |
| Answer» B. \[{{e}^{2}}-1\] | |
| 1056. |
The value of \[\int{{{e}^{ta{{n}^{-1}}}}^{x}\frac{(1+x+{{x}^{2}})}{1+{{x}^{2}}}dx}\] is |
| A. | \[x{{e}^{{{\tan }^{-1}}}}x+c\] |
| B. | \[{{\tan }^{-1}}x+C\] |
| C. | \[{{e}^{{{\tan }^{-1}}x}}+2x+C\] |
| D. | None of these |
| Answer» B. \[{{\tan }^{-1}}x+C\] | |
| 1057. |
If \[\int\limits_{0}^{\infty }{{{e}^{-ax}}dx=\frac{1}{a},}\] then \[\int\limits_{0}^{\infty }{{{x}^{n}}{{e}^{-ax}}dx}\] is |
| A. | \[\frac{{{(-1)}^{n}}n!}{{{a}^{n+1}}}\] |
| B. | \[\frac{{{(-1)}^{n}}(n-1)!}{{{a}^{n}}}\] |
| C. | \[\frac{n!}{{{a}^{n+1}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1058. |
If \[\phi (x)=\int{{{\cot }^{4}}xdx+\frac{1}{3}{{\cot }^{3}}x-\cot x}\] and\[\phi \left( \frac{\pi }{2} \right)=\frac{\pi }{2}\] then \[\phi (x)\] is |
| A. | \[\pi -x\] |
| B. | \[x-\pi \] |
| C. | \[\pi /2-x\] |
| D. | x |
| Answer» E. | |
| 1059. |
\[\int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\frac{\left| x \right|dx}{8{{\cos }^{2}}2x+1}}\] has the value |
| A. | \[\frac{{{\pi }^{2}}}{6}\] |
| B. | \[\frac{{{\pi }^{2}}}{12}\] |
| C. | \[\frac{{{\pi }^{2}}}{24}\] |
| D. | None of these |
| Answer» C. \[\frac{{{\pi }^{2}}}{24}\] | |
| 1060. |
\[\int{\frac{{{x}^{n-1}}}{{{x}^{2n}}+{{a}^{2}}}dx}=\] |
| A. | \[\frac{1}{na}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] |
| B. | \[\frac{n}{a}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] |
| C. | \[\frac{n}{a}{{\sin }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] |
| D. | \[\frac{n}{a}{{\cos }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] |
| Answer» B. \[\frac{n}{a}{{\tan }^{-1}}\left( \frac{{{x}^{n}}}{a} \right)+C\] | |
| 1061. |
If \[\int{\frac{{{x}^{2}}-x+1}{{{x}^{2}}+1}{{e}^{{{\cot }^{-1}}x}}dx=A(x){{e}^{{{\cot }^{-1}}x}}+C}\], then \[A(x)\] is equal to: |
| A. | \[-x\] |
| B. | \[x\] |
| C. | \[\sqrt{1-x}\] |
| D. | \[\sqrt{1+x}\] |
| Answer» C. \[\sqrt{1-x}\] | |
| 1062. |
\[\int\limits_{0}^{1}{\frac{1}{\left( {{x}^{2}}+16 \right)\left( {{x}^{2}}+25 \right)}dx=}\] |
| A. | \[\frac{1}{5}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] |
| B. | \[\frac{1}{9}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] |
| C. | \[\frac{1}{4}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] |
| D. | \[\frac{1}{9}\left[ \frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] |
| Answer» C. \[\frac{1}{4}\left[ \frac{1}{4}{{\tan }^{-1}}\left( \frac{1}{4} \right)-\frac{1}{5}{{\tan }^{-1}}\left( \frac{1}{5} \right) \right]\] | |
| 1063. |
If \[f(x)=A\,\,\sin \left( \frac{\pi x}{2} \right)+B\] and \[f'\left( \frac{1}{2} \right)=\sqrt{2}\] and \[\int_{0}^{1}{f(x)dx=\frac{2A}{\pi }}\], then what is the value of B? |
| A. | \[\frac{2}{\pi }\] |
| B. | \[\frac{4}{\pi }\] |
| C. | 0 |
| D. | 1 |
| Answer» D. 1 | |
| 1064. |
What is \[\int\limits_{0}^{1}{\frac{{{\tan }^{-1}}}{1+{{x}^{2}}}dx}\] equal to? |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{8}\] |
| C. | \[\frac{{{\pi }^{2}}}{8}\] |
| D. | \[\frac{{{\pi }^{2}}}{32}\] |
| Answer» E. | |
| 1065. |
\[\int{{{\left( x+\frac{1}{x} \right)}^{n+5}}\left( \frac{{{x}^{2}}-1}{{{x}^{2}}} \right)dx}\] is equal to: |
| A. | \[\frac{{{\left( x+\frac{1}{x} \right)}^{n+6}}}{n+6}+c\] |
| B. | \[{{\left[ \frac{{{x}^{2}}+1}{{{x}^{2}}} \right]}^{n+6}}(n+6)+c\] |
| C. | \[{{\left[ \frac{x}{{{x}^{2}}+1} \right]}^{n+6}}(n+6)+c\] |
| D. | None of these |
| Answer» B. \[{{\left[ \frac{{{x}^{2}}+1}{{{x}^{2}}} \right]}^{n+6}}(n+6)+c\] | |
| 1066. |
If \[\int{\frac{dx}{{{x}^{22}}({{x}^{7}}-6)}}\]\[=A\{In{{(p)}^{6}}+9{{p}^{2}}-2{{p}^{3}}-18p\}+c\] then |
| A. | \[A=\frac{1}{9072},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\] |
| B. | \[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)\] |
| C. | \[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}}{{{x}^{7}}-6} \right)\] |
| D. | \[A=\frac{1}{9072},p={{\left( \frac{{{x}^{7}}-6}{{{x}^{7}}} \right)}^{-1}}\] |
| Answer» C. \[A=\frac{1}{54432},p=\left( \frac{{{x}^{7}}}{{{x}^{7}}-6} \right)\] | |
| 1067. |
The value of \[\int{\frac{\sin x}{\sin 4x}dx}\] is |
| A. | \[\frac{1}{4}\log \left| \frac{\sin x-1}{\sin x+1} \right|-\frac{1}{\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\] |
| B. | \[\frac{1}{8}\log \left| \frac{\cos x-1}{\cos x+1} \right|-\frac{1}{2\sqrt{2}}\log \left| \frac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1} \right|+C\] |
| C. | \[\frac{1}{8}\log \left| \frac{\sin x-1}{sinx+1} \right|-\frac{1}{4\sqrt{2}}\log \left| \frac{\sqrt{2}\sin x-1}{\sqrt{2}\sin x+1} \right|+C\] |
| D. | None of these. |
| Answer» D. None of these. | |
| 1068. |
If\[I=\int{{{\sin }^{-\frac{11}{3}}}x{{\cos }^{-\frac{1}{3}}}xdx}\]\[=A{{\cot }^{2/3}}x+B{{\cot }^{8/3}}x+C\]. Then |
| A. | \[A=\frac{2}{3},B=\frac{8}{3}\] |
| B. | \[A=-\frac{3}{2},B=-\frac{3}{8}\] |
| C. | \[A=\frac{3}{2},B=\frac{3}{8}\] |
| D. | None of these |
| Answer» C. \[A=\frac{3}{2},B=\frac{3}{8}\] | |
| 1069. |
If \[f(x)\] and \[\phi (x)\] are continuous functions on the interval \[[0,4]\] satisfying \[f(x)=f(4-x)\], \[\phi (x)+\phi (4-x)=3\] and \[\int\limits_{0}^{4}{f(x)dx=2,}\] then \[\int\limits_{0}^{4}{f(x)\phi (x)dx}\] |
| A. | 3 |
| B. | 6 |
| C. | 2 |
| D. | None of these |
| Answer» B. 6 | |
| 1070. |
What is \[\int{\frac{\log x}{{{(1+\log \,x)}^{2}}}dx}\] equal to? |
| A. | \[\frac{1}{{{\left( 1+\log x \right)}^{3}}}+c\] |
| B. | \[\frac{1}{{{\left( 1+\log x \right)}^{2}}}+c\] |
| C. | \[\frac{x}{\left( 1+\log x \right)}+c\] |
| D. | \[\frac{x}{{{\left( 1+\log x \right)}^{2}}}+c\] Where c is a constant. |
| Answer» D. \[\frac{x}{{{\left( 1+\log x \right)}^{2}}}+c\] Where c is a constant. | |
| 1071. |
What is \[\int{\sin x\log (\tan x)dx}\] equal to? |
| A. | \[\cos x\log \tan x+\log \,\,\tan (x/2)+c\] |
| B. | \[-\cos x\log \tan x+\log \,\,\tan (x/2)+c\] |
| C. | \[\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\] |
| D. | \[-\cos x\log \tan x+\log \,\,\cot \,(x/2)+c\] |
| Answer» B. \[-\cos x\log \tan x+\log \,\,\tan (x/2)+c\] | |
| 1072. |
If m is an integer, then \[\int_{0}^{\pi }{\frac{\sin (2mx)}{\sin x}dx}\] is equal to: |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | \[\pi \] |
| Answer» D. \[\pi \] | |
| 1073. |
\[\int{\sin 2x.\log \cos x\,\,dx}\] is equal to: |
| A. | \[{{\cos }^{2}}x\left( \frac{1}{2}+\log \cos x \right)+k\] |
| B. | \[{{\cos }^{2}}x.\log \,\,\cos \,\,x+k\] |
| C. | \[{{\cos }^{2}}x\left( \frac{1}{2}-\log \cos x \right)+k\] |
| D. | None of these |
| Answer» D. None of these | |
| 1074. |
\[\int{{{e}^{3\log x}}{{({{x}^{4}}+1)}^{-1}}dx}\] is equal to |
| 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. \[-\log ({{x}^{4}}+1)+C\] | |
| 1075. |
The value of \[\int\limits_{0}^{1}{\frac{dx}{{{e}^{x}}+e}}\] is equal to |
| A. | \[\frac{1}{e}\log \left( \frac{1+e}{2} \right)\] |
| B. | \[\log \left( \frac{1+e}{2} \right)\] |
| C. | \[\frac{1}{e}\log (1+e)\] |
| D. | \[\log \left( \frac{2}{1+e} \right)\] |
| Answer» B. \[\log \left( \frac{1+e}{2} \right)\] | |
| 1076. |
Let \[f:R\to R\] is differentiable function and \[f(1)=4,\] then the value of \[\underset{x\to 1}{\mathop{\lim }}\,\int\limits_{0}^{f(x)}{\frac{2tdt}{x-1}}\] is |
| A. | \[8f'(1)\] |
| B. | \[4f'(1)\] |
| C. | \[2f'(1)\] |
| D. | \[f'(1)\] |
| Answer» B. \[4f'(1)\] | |
| 1077. |
\[\int{\frac{x-1}{{{(x+1)}^{2}}\sqrt{{{x}^{3}}+{{x}^{2}}+x}}dx}\] is equal to |
| A. | \[{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] |
| B. | \[2{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] |
| C. | \[3{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] |
| D. | None of these |
| Answer» C. \[3{{\tan }^{-1}}\sqrt{\frac{{{x}^{2}}+x+1}{x}}+C\] | |
| 1078. |
What is \[\int\limits_{0}^{\pi /2}{\sin \,\,2x\,\,\ell n\,(\cot \,\,x)dx}\] equal to? |
| A. | 0 |
| B. | \[\pi \ell n2\] |
| C. | \[-\pi \ell n2\] |
| D. | \[\frac{\pi \ell n2}{2}\] |
| Answer» B. \[\pi \ell n2\] | |
| 1079. |
What is \[\int{{{\tan }^{2}}x{{\sec }^{4}}x\,dx}\] equal to? |
| A. | \[\frac{{{\sec }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\] |
| B. | \[\frac{{{\tan }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\] |
| C. | \[\frac{{{\tan }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\] |
| D. | \[\frac{{{\sec }^{5}}x}{5}+\frac{{{\tan }^{3}}x}{3}+c\] |
| Answer» C. \[\frac{{{\tan }^{5}}x}{5}+\frac{{{\sec }^{3}}x}{3}+c\] | |
| 1080. |
Let \[f:R\to R\] and \[g:R\to R\] be continuous functions. Then the value of \[\int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\{f(x)+f(-x)\}\{g(x)-g(-x)\}dx}\] is |
| A. | \[f(x)g(x)\] |
| B. | \[f(x)+g(x)\] |
| C. | 0 |
| D. | None of theses |
| Answer» D. None of theses | |
| 1081. |
If \[\int{\frac{1}{1+\sin x}dx=\tan \left( \frac{x}{2}+a \right)+b}\] then |
| A. | \[a=-\frac{\pi }{4},b\in R\] |
| B. | \[a=\frac{\pi }{4},b\in R\] |
| C. | \[a=\frac{5\pi }{4},b\in R\] |
| D. | None of these |
| Answer» B. \[a=\frac{\pi }{4},b\in R\] | |
| 1082. |
Let \[f:(0,\infty )\to R\] and \[F(x)=\int\limits_{0}^{x}{f(t)dt}\].If \[F({{x}^{2}})={{x}^{2}}(1+x),\] then \[f(4)\] equals |
| A. | \[\frac{5}{4}\] |
| B. | 7 |
| C. | 4 |
| D. | 2 |
| Answer» D. 2 | |
| 1083. |
If \[\int{\frac{x{{e}^{x}}}{\sqrt{1+{{e}^{x}}}}dx=f(x)\sqrt{1+{{e}^{x}}}-2\log \,\,g(x)+C,}\] then |
| A. | \[f(x)=x-1\] |
| B. | \[g(x)=\frac{\sqrt{1+{{e}^{x}}}-1}{\sqrt{1+{{e}^{x}}}+1}\] |
| C. | \[g(x)=\frac{\sqrt{1+{{e}^{x}}}+1}{\sqrt{1+{{e}^{x}}}-1}\] |
| D. | \[f(x)=2(2-x)\] |
| Answer» C. \[g(x)=\frac{\sqrt{1+{{e}^{x}}}+1}{\sqrt{1+{{e}^{x}}}-1}\] | |
| 1084. |
If \[A=\int\limits_{0}^{1}{\frac{{{e}^{t}}}{t+1}dt,}\] then \[\int\limits_{0}^{1}{{{e}^{t}}\log (1+t)dt}\] in terms of A equals |
| A. | \[e\log (A)\] |
| B. | \[\frac{e}{2}-A\] |
| C. | \[e-l-\frac{A}{2}\] |
| D. | \[\frac{e}{2}-l-A\] |
| Answer» E. | |
| 1085. |
The tangent of the curve \[y=f(x)\] at the point with abscissa \[x=1\] from an angle of \[\pi /6\] and at the point \[x=2\] an angle of \[\pi /3\] and at the point \[x=3\] an angle of \[\pi /4\]. If \[f''(x)\] is continuous, then the value of \[\int\limits_{1}^{3}{f''(x)f'(x)dx+\int\limits_{2}^{3}{f''(x)dx}}\] is |
| A. | \[\frac{4\sqrt{3}-1}{3\sqrt{3}}\] |
| B. | \[\frac{3\sqrt{3}-1}{2}\] |
| C. | \[\frac{4-3\sqrt{3}}{3}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1086. |
Let\[f(x)=\int{{{e}^{x}}(x-1)(x-2)}dx\]. Then f decreases in the interval |
| A. | \[(-\infty ,-2)\] |
| B. | \[(-2,-1)\] |
| C. | \[(1,\,\,2)\] |
| D. | \[(2,+\infty )\] |
| Answer» D. \[(2,+\infty )\] | |
| 1087. |
If \[I=\int{\frac{1}{2p}\sqrt{\frac{p-1}{p+1}}dp=f(p)+c}\], then f(p) is equal to: |
| A. | \[\frac{1}{2}\ell n\left[ p-\sqrt{{{p}^{2}}-1} \right]\] |
| B. | \[\frac{1}{2}{{\cos }^{-1}}p+\frac{1}{2}{{\sec }^{-1}}p\] |
| C. | \[\ell n\sqrt{p+\sqrt{{{p}^{2}}-1}}-\frac{1}{2}{{\sec }^{-1}}p\] |
| D. | None of the above. |
| Answer» D. None of the above. | |
| 1088. |
\[\int{\frac{\{f(x).\phi '(x)-f'(x).\phi (x)\}}{f(x).\phi (x)}}\log \frac{f(x)}{\phi (x)}dx\] is equal to: |
| A. | \[\log \frac{\phi (x)}{f(x)}+k\] |
| B. | \[\frac{1}{2}{{\left\{ \log \frac{\phi (x)}{f(x)} \right\}}^{2}}+k\] |
| C. | \[\frac{\phi (x)}{f(x)}\log \frac{\phi (x)}{f(x)}+k\] |
| D. | None of these |
| Answer» C. \[\frac{\phi (x)}{f(x)}\log \frac{\phi (x)}{f(x)}+k\] | |
| 1089. |
What is \[\int{{{e}^{ln\,\,x}}\sin x\,\,dx}\] equal to? |
| A. | \[{{e}^{ln\,\,x}}(\sin \,\,x-\cos \,\,x)+c\] |
| B. | \[(\sin \,\,x-x\,\,\cos \,\,x)+c\] |
| C. | \[(x\,\,\sin \,\,x+\cos \,\,x)+c\] |
| D. | \[(\sin \,\,x+x\,\,\cos \,\,x)-c\] Where ?c? is a constant of integration. |
| Answer» C. \[(x\,\,\sin \,\,x+\cos \,\,x)+c\] | |
| 1090. |
If\[\int{\frac{dx}{f(x)}=\log {{\{f(x)\}}^{2}}+c}\], then what is \[f(x)\] equal to? |
| A. | \[2x+\alpha \] |
| B. | \[x+\alpha \] |
| C. | \[\frac{x}{2}+\alpha \] |
| D. | \[{{x}^{2}}+\alpha \] |
| Answer» D. \[{{x}^{2}}+\alpha \] | |
| 1091. |
\[\int\limits_{0}^{\infty }{\frac{dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})}}\] is |
| A. | \[\frac{\pi ab}{a+b}\] |
| B. | \[\frac{\pi }{2(a+b)}\] |
| C. | \[\frac{\pi }{2ab(a+b)}\] |
| D. | \[\frac{\pi (a+b)}{2ab}\] |
| Answer» D. \[\frac{\pi (a+b)}{2ab}\] | |
| 1092. |
\[\int_{0}^{1}{[f(x)g''(x)-f''(x)g(x)]dx}\] is equal to: [Given f(0) = g (0) = 0] |
| A. | \[f(1)g(1)-f(1)g'(1)\] |
| B. | \[f(1)g'(1)+f'(1)g(1)\] |
| C. | \[f(1)g'(1)-f'(1)g(1)\] |
| D. | None of these |
| Answer» D. None of these | |
| 1093. |
If \[f(p,q)=\int_{0}^{\pi /2}{{{\cos }^{p}}x\cos \,\,qx\,\,dx}\], then |
| A. | \[f(p,q)=\frac{q}{p+q}f(p-1,q-1)\] |
| B. | \[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\] |
| C. | \[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\] |
| D. | \[f(p,q)=-\frac{q}{p+q}f(p-1,q-1)\] |
| Answer» C. \[f(p,q)=\frac{p}{p+q}f(p-1,q-1)\] | |
| 1094. |
If \[\int\limits_{-3}^{2}{f(x)dx=\frac{7}{3}}\] and \[\int\limits_{-3}^{9}{f(x)dx=-\frac{5}{6}}\], then what is the value of \[\int\limits_{2}^{9}{f(x)dx?}\] |
| A. | \[\frac{-19}{6}\] |
| B. | \[\frac{19}{6}\] |
| C. | \[\frac{3}{2}\] |
| D. | \[-\frac{3}{2}\] |
| Answer» B. \[\frac{19}{6}\] | |
| 1095. |
What is \[\int\limits_{-\frac{\pi }{6}}^{\frac{\pi }{6}}{\frac{{{\sin }^{5}}x{{\cos }^{3}}x}{{{x}^{4}}}}dx\] equal to? |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{8}\] |
| D. | 0 |
| Answer» E. | |
| 1096. |
The function \[f(x)=\int\limits_{-1}^{x}{t({{e}^{t}}-1)(t-1){{(t-2)}^{3}}{{(t-3)}^{5}}}\] dt has a local minimum at x = |
| A. | 0 |
| B. | 1, 3 |
| C. | 2 |
| D. | None of these |
| Answer» C. 2 | |
| 1097. |
If \[\int{f(x)\cos \,\,x\,\,dx=\frac{1}{2}{{f}^{2}}(x)+c,}\] then \[f(x)\] can be |
| A. | x |
| B. | 1 |
| C. | \[\cos x\] |
| D. | \[sinx\] |
| Answer» E. | |
| 1098. |
The line \[y=\alpha \] intersects the curve \[y=g(x)\],atleast at two points. If \[\int\limits_{2}^{x}{g(t)dt=\frac{{{x}^{2}}}{2}+\int\limits_{x}^{2}{{{t}^{2}}g(t)dt}}\]then possible value of \[\alpha \] is/are- |
| A. | \[\left( -\frac{1}{2},\frac{1}{2} \right)\] |
| B. | \[\left[ -\frac{1}{2},\frac{1}{2} \right]\] |
| C. | \[\left( -\frac{1}{2},\frac{1}{2} \right)-\{0\}\] |
| D. | \[\left\{ -\frac{1}{2},0,\frac{1}{2} \right\}\] |
| Answer» D. \[\left\{ -\frac{1}{2},0,\frac{1}{2} \right\}\] | |
| 1099. |
\[\int{{{x}^{51}}({{\tan }^{-1}}x+{{\cot }^{-1}}x)dx}\] |
| A. | \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\] |
| B. | \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\] |
| C. | \[\frac{\pi {{x}^{52}}}{104}+\frac{\pi }{2}+c\] |
| D. | \[\frac{{{x}^{52}}}{52}+\frac{\pi }{2}+c\] |
| Answer» B. \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\] | |
| 1100. |
\[\int{\frac{dx}{\cos x+\sqrt{3}\sin x}}\] equals |
| A. | \[\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\] |
| B. | \[\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\] |
| C. | \[\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\] |
| D. | \[\frac{1}{2}\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\] |
| Answer» D. \[\frac{1}{2}\log \tan \left( \frac{x}{2}-\frac{\pi }{12} \right)+C\] | |