#### Answer

The methods are: substitution, integration by parts, integration by parts, substitution.

#### Work Step by Step

For $\int x \cos (x^2)dx $, we can use the substitution method.
For $\int x \cos (x)dx $, we can use the Integration by Parts method.
For $\int x^2 e^x dx $, we can use the Integration by Parts method.
For $\int x^2 e^x dx $, we can use the substitution method.