Hey!
Can anyone give me good examples on how to solve integrals like:
$\displaystyle \int_{0}^{\pi/2} \frac{cos x}{sin^2x + sin^3 x} dx$
with partial integration/substitution methods?
First rewrite the integrand as follows
$\displaystyle \int_0^{\pi/2}\frac{\cos x}{\sin^2x+\sin^3x}\,dx=\int_0^{\pi/2}\frac{\cos x}{\sin^2x(1+\sin x)}\,dx$
Then set $\displaystyle u=1+\sin x\implies du=\cos x\,dx$, which yields
$\displaystyle \int_1^2\frac1{u(u-1)^2}\,du$
Just post if you cannot take it from there.
This step is not correct 'cause "u" it's a function of "x", not a constant.
We have
$\displaystyle
\begin{aligned}
\frac1{u(u-1)^2}&=\frac{u^2-2u+1+u-u^2+u}{u(u-1)^2}\\&=\frac{(u-1)^2+u-u(u-1)}{u(u-1)^2}\\&=\frac1u+\frac1{(u-1)^2}-\frac1{u-1}
\end{aligned}
$
Kill it now.