In this problem you will compute the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}2sin(x)cos(x)dx$ in three different ways.

$($a$)$ Compute the integral using the substitution $u=sin(x).$

$($b$)$ Compute the integral using the substitution $u=cos(x).$

$($c$)$ Compute the integral by applying the trig identity $2sin(x)cos(x)=sin(2x)$ and then using the substitution $u=2x.$

$($d$)$ Use the linked graph to explain why all three answers to parts $($a$),($b$),($c$)$ are correct even though they look different.

$($e$)$ Compute ${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}2sin(x)cos(x)dx$ three different times, using your answers to parts $($a$),($b$),$ and $($c$),$ and check that all three computations give the same result.