Your turn

(Friday, August 27, 2021
Revised August 29, 2021)

1)

$\int x^{5}\sin(x^{3})~{}dx$
2)

$\int_{0}^{2\pi}\sin^{2}(x)~{}dx$

In class we did this without the limits. It may look like one where you integrate by parts twice. However, if you do that, you get $0=0$ ! The trig identity $\cos^{2}(x)+\sin^{2}(x)=1$ is very useful in this problem.
3)

$\int\arctan(x)~{}dx$

Again, key is to see this as $\int 1\cdot\arctan(x)~{}dx$ .