$(16$ points$)$ Evaluate the following definite integrals.

a$.{\displaystyle \underset{0}{\overset{2}{}}}(4+2t+3t^2)dt$

b$.{\displaystyle \underset{1}{\overset{3}{}}}\frac{v^3+3v^6}{v^4}dv$

c$.{\displaystyle \underset{\frac{1}{\sqrt[\phantom{2}]{3}}}{\overset{\sqrt[\phantom{2}]{3}}{}}}\frac{6}{1+x^2}dx$

d$.{\displaystyle \underset{e}{\overset{e^3}{}}}\frac{e^{\frac{1}{x}}dx}{x^2}$