Compute the following integrals. Use proper notation; show work.

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{x^2+1}dx=$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{e^{\sqrt[\phantom{2}]{x}}}{2\sqrt[\phantom{2}]{x}}dx=$

$($d$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{4}{5+x}dx=$

$($e$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{6x}{9+x^2}dx=$

$($f$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x}{\sqrt[\phantom{2}]{6+x^2}}dx=$