$(b)$ Evaluate the integral

${\displaystyle \underset{1}{\overset{}{}}}\frac{2}{(x+1{)}^5}dx$

$or$ show that $it$ diverges.

$(c)$ Consider the power series

${\displaystyle \underset{k=0}{\overset{}{}}}\frac{2^k(x^2+1{)}^k}{k!}$

Show that the series converges for every xinR.

$(d)$ Consider the integral

${\displaystyle \underset{1}{\overset{}{}}}\frac{1}{x^6+5}dx$

Show that $it$ converges, $by$ a suitable comparison $or$ otherwise.