integrals with $u-$substitution method:

$($d$){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}co{s}^2(2x)sin(2x)dx.$

$($e$){\displaystyle \underset{0}{\overset{1}{}}}{x}^4(x^5-1{)}^{20}dx$

$($f$){\displaystyle \underset{0}{\overset{2}{}}}\frac{t^2}{\sqrt[\phantom{2}]{t^3+1}}dt.$