Review of Integration II (Rational functions and Universal Substitutions)

Q. The solution for ${\int }_0^{\pi /6}co{s}^{4}3\theta si{n}^{3}6\theta d\theta$ is

1. $0$
2. $1/15$
3. $1$
4. $8/3$

Q. If ${\int }_0^{2\pi }\left|xsinx\right|dx=k\pi ,$ then the values of $k$ is equal to

1. 4

Q. The value of the integral ${\int }_{-\infty }^{\infty }12cos\left(2\pi t\right)\frac{sin\left(4\pi t\right)}{4\pi t}dt$ is

1. 3

Q. The integral $\frac{1}{2\pi }{\int }_0^{2\pi }sin(t-\tau )cos\tau d\tau$ equals

1. $sintcost$
2. $0$
3. $\left(1/2\right)cost$
4. $\left(1/2\right)sint$

Q. Consider the following definite integral:
$I={\int }_0^1\frac{(si{n}^{-1}x{)}^2}{\sqrt{1-x^2}}dx$
The value of the integral is

1. $\frac{{\pi }^3}{24}$
2. $\frac{{\pi }^3}{12}$
3. $\frac{{\pi }^3}{48}$
4. $\frac{{\pi }^3}{64}$

Q. Let $x$ be a continous variable defined over the interval $(-\infty ,\infty )$ and $f\left(x\right)=e^{-x-e^{-x}}$. The integral $g\left(x\right)=\int f\left(x\right)dx$ is equal to

1. $e^{e^{-x}}$
2. $e^{-e^{-x}}$
3. $e^{-e^x}$
4. $e^{-x}$

Q. The integral ${\int }_0^1\frac{dx}{\sqrt{(1-x)}}$ is equal to

1. 2

Q. $\text{If}\int e^x{\left(\frac{1-x}{1+x^2}\right)}^{2}dx=\frac{e^{xc}}{a+b{x}^2}+k\text{, then the sum of a, b and c is \_\_\_.}$

1. 3

Q. The value of the integral given below is ${\int }_0^{\pi }{x}^{2}cosxdx$

1. $-2\pi$
2. $\pi$
3. $-\pi$
4. $2\pi$

Q. The value of ${\int }_0^{\infty }\frac{1}{1+x^2}dx+{\int }_0^{\infty }\frac{sinx}{x}dx$ is

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

Q. The value of the integral ${\int }_0^2\frac{(x-1{)}^{2}sin(x-1)}{(x-1{)}^2+cos(x-1)}dx$ is

1. $3$
2. $0$
3. $-1$
4. $-2$