Integral between 0 and pi/2 of cos(x)sin^2(x)

The key point in here is noticing that the derivative of sin(x) is cos(x). That way, if we rename sin(x) as u , i.e u:=sin(x), the derivative will be du=cos(x)dx. So, inside the integral, instead of having cos(x)sin^2(x)dx we can have u^2du. This tecnique is called a change of variables. However, it is not completed since once we've changed the variable from x to u we need to change the limits of integration as well. This means that when x=0 then u=0 and when x=pi/2 then u=1 (check the graph of the sin function to guide yourself). This way we write:integral from 0 to pi/2 of cos(x)sin^2(x)dx= integral from 0 to 1 of u^2du= 1/3 of u^3 evaluated from 0 to 1= 1/31^3-1/30^3=1/3.The second equality is satisfied because the primitive of the polinomyal function u^2 is 1/3u ^3 since the derivative of 1/3u ^3 is u^2.