$[1]$ Evaluate the following integrals:

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

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{co{s}^{3}x}{1-sinx}dx$

$[2]$ a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}arctan2xdx$

$($b$)$ Evaluate the improper integral ${\displaystyle \underset{0}{\overset{}{}}}\frac{xdx}{1+x^4}$