Tricky examples

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

1)

$\int x^{3}e^{x^{2}}~{}dx$

(This is from last time.) There are two choices: substituion then integrate by parts, or integrate by parts then sub.
2)

$\int e^{x}\cos(x)~{}dx$

Note: this is a standard example where we integrate by parts twice and do a bit of algebra to find an answer.
3)

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

Key is to see this as $\int\ln(x)~{}dx=\int 1\cdot\ln(x)~{}dx$ …