Integration Using Substitution

Q. The integral $\underset{\pi /6}{\overset{\pi /4}{\int }}\frac{dx}{\sin 2x({\tan }^{5}x+{\cot }^{5}x)}$ equals :

1. $\frac{1}{10}[\frac{\pi }{4}-{\tan }^{-1}\left(\frac{1}{9\sqrt{3}}\right)]$
2. $\frac{1}{5}[\frac{\pi }{4}-{\tan }^{-1}\left(\frac{1}{3\sqrt{3}}\right)]$
3. $\frac{\pi }{40}$
4. $\frac{1}{20}\left[{\tan }^{-1}\left(\frac{1}{9\sqrt{3}}\right)\right]$

Q. Let $f$ be a real-valued function and satisfies $f\left(x\right)+f\left(y\right)=\frac{1}{x}+\frac{1}{y}\forall x,y\in R-\left\{0\right\}.$ If $\underset{2}{\overset{3}{\int }}\left(\frac{3(f\left(x\right){)}^5-f\left(x\right)}{1-(f\left(x\right){)}^4}\right)dx=\frac{1}{2}\ln \left(\frac{2^{\alpha }}{3^{\beta }}\right),$ then

1. $\alpha \le \beta$
2. $\alpha +\beta =17$
3. ${\tan }^{-1}\sqrt{\alpha -\beta }={\sec }^{-1}\left(\frac{\alpha }{5}\right)$
4. $(\alpha -\beta )$ is not a prime number

Q. The value of ${\int }_0^{\pi }\frac{1}{5+4\cos x}\mathrm{dx}$ is

1. $\pi$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. $\text{For}x>0,\text{let}f\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{\log t}{1+t}dt.\text{Then}f\left(x\right)+f\left(\frac{1}{x}\right)\text{is equal to}$

1. $\frac{1}{4}\log x^2$
2. $\frac{1}{4}(\log x{)}^2$
3. $\log x$
4. $\frac{1}{2}(\log x{)}^2$

Q. Let $f\left(x\right)={\sin }^{-1}x$ and $g\left(x\right)=\frac{x^2-x-2}{2x^2-x-6}$. If $g\left(2\right)=\underset{x\to 2}{lim}g\left(x\right)$, then the domain of the function $fog$ is :

1. $(-\infty ,-1]\cup [2,\infty )$
2. $(-\infty ,-2]\cup [-\frac{4}{3},\infty )$
3. $(-\infty ,-2]\cup [-1,\infty )$
4. $(-\infty ,-2]\cup [-\frac{3}{2},\infty )$

Q. If $\underset{\ln 2}{\overset{k}{\int }}\frac{1}{\sqrt{e^x-1}}dx=\frac{\pi }{6},$ then $k=\ln p.$ The value of $p+1$ is

Q. $\text{For}x>0,\text{let}f\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{\log t}{1+t}dt.\text{Then}f\left(x\right)+f\left(\frac{1}{x}\right)\text{is equal to}$

1. $\frac{1}{4}\log x^2$
2. $\frac{1}{4}(\log x{)}^2$
3. $\log x$
4. $\frac{1}{2}(\log x{)}^2$

Q. If $y=\underset{x^2}{\overset{x^3}{\int }}\frac{1}{\ln t}dt$ $(\text{where}x\in R^{+}-\{1\left\}\right)$, then the value of $\frac{dy}{dx}$ is

1. $x(x-1)(\ln x{)}^{-1}$
2. $x(x-1)(\ln x{)}^2$
3. $x(x-1)\left(\ln x\right)$
4. $x(x-1)(\ln x{)}^{-2}$

Q. Let $f:[0,27]\to [\frac{1}{3},6]$ be a differentiable function such that $f^{\prime }\left(x\right)<0\forall x\in D_f$. If $\underset{0}{\overset{27}{\int }}x{f}^{\prime }\left(x\right)dx=\lambda -3\underset{0}{\overset{3}{\int }}{x}^{2}f\left(x^3\right)dx,$ then the minimum value of $\lambda$ is
(Note: $D_f$ denotes the domain of the function)