Review of Integration I (Rational functions and Substitutions)

Q. The value of the integral $I=\frac{1}{\sqrt{2\pi }}{\int }_0^{\infty }exp(-\frac{x^2}{8})dx$ is

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

Q.

If $y={\tan }^{-1}\left(\frac{x^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}+a^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{1-x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{a}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)$, then $\frac{dy}{dx}=$

1. $\frac{1}{3x^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}$

2. $\frac{a}{3x^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}$

3. $-\frac{1}{3x^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}$

4. $-\frac{a}{3x^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}$

Q.

What is the $arcsin$?

Q. The value of the integral ${\int }_{-\infty }^{\infty }\frac{dx}{1+x^2}$ is

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

Q.

Can an integral have two answers?

Q. The value of the integral ${\int }_{-1}^1\frac{1}{x^2}dx$

1. $2$
2. does not exist
3. $-2$
4. $\infty$