Evaluate the following integrals:

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^2(3x)dx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{co{s}^2(lnt)}{t}dt$

Evaluate the following integrals:

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{sinx}co{s}^{3}xdx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^{2}tsin2tdt$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cotxco{s}^{2}xdx$

Evaluate the following integrals using trigonometric substitution.

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{x^2\sqrt[\phantom{2}]{4-x^2}}$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^2\sqrt[\phantom{2}]{3+2x-x^2}dx$

Evaluate the following integrals:

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{3x-2}{x+1}dx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{10}{(y-1)(y^2+9)}dy$

Determine whether the following integrals converge or diverge.

$($a$){\displaystyle \underset{0}{\overset{4}{}}}\frac{dt}{t^2-t-2}$

$($b$){\displaystyle \underset{-}{\overset{}{}}}x{e}^{-x2}$dx