Integration Using Substitution

Q. Consider the equation
$\underset{1}{\overset{e}{\int }}\frac{({\log }_{e}x{)}^{1/2}}{x(a-({\log }_{e}x{)}^{3/2}{)}^2}dx,a\in (-\infty ,0)\cup (1,\infty )$
Which of the following statements is/are TRUE?

1. More than one $a$ satisfy the above equation
2. An irrational number $a$ satisfies the above equation
3. An integer $a$ satisfies the above equation
4. No $a$ satisfies the above equation

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. If $I_n={\int }_0^{\frac{\pi }{4}}{\tan }^{n}x\mathrm{dx}$, then the value of $I_9+I_{11}$ is

1. $\frac{1}{9}$
2. $\frac{1}{10}$
3. $\frac{1}{11}$
4. $\frac{1}{8}$

Q. $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}{\log }_e(\sin x+\cos x)dx$ is

1. $-\frac{\pi }{4}\ln 2$
2. $\frac{\pi }{4}\ln 2$
3. $\frac{\pi }{8}\ln 2$
4. $-\frac{\pi }{8}\ln 2$

Q. If $\underset{0}{\overset{\pi /3}{\int }}\frac{\tan \theta }{\sqrt{2k\sec \theta }}d\theta =1-\frac{1}{\sqrt{2}},(k>0)$, then the value of $k$ is :

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

Q. The integral $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx$ equals to:

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

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

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 $f:R\to R$ be a function defined as $f\left(x\right)=(x-1)(x+2)(x-3)(x-6)-100.$ If $g\left(x\right)$ is a polynomial of degree at most $3$ such that $\int \frac{g\left(x\right)}{f\left(x\right)}dx$ does not contain any logarithmic function and $g(-2)=-10,$ then

1. Minimum value of $f\left(x\right)$ is $-84$ when $x=2$
2. Minimum value of $f\left(x\right)$ is $-42$ when $x=2$
3. $\underset{0}{\overset{4}{\int }}\frac{g\left(x\right)}{f\left(x\right)}dx=\frac{\pi }{2}$
4. $\underset{0}{\overset{4}{\int }}\frac{g\left(x\right)}{f\left(x\right)}dx=\pi$

Q. If $I_n=\underset{0}{\overset{1}{\int }}({\cos }^{-1}x{)}^{n}dx,$ then $I_4$ is equal to

1. $\frac{{\pi }^3}{2}-12\pi -24$
2. $\frac{{\pi }^3}{2}-12\pi +24$
3. $\frac{{\pi }^3}{2}-12\pi$
4. $\frac{{\pi }^3}{4}-12\pi +24$