Evaluate the definite integral: ${\displaystyle \underset{0}{\overset{1}{}}}{x}^3{e}^{x}dx$

First make a substitution then use integration by parts to evaluate the integral.

a$.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cos(ln(x))dx$

b$.{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{3}}{}}}arctan(\frac{1}{x})dx$

Use the Method of Cylindrical Shells to set up and evaluate the volume of the solid obtained by rotating the region bounded between $y=sin(x),y=0,x=0$ and $x=$ about the $y-$axis.