Integration by substitution is basically the chain rule in reverse.

For example, because

we know that

More generally, because

we can say that

However, sometimes it is very hard to see that a function is the result of a chain rule, and so we use something called $u$-substitution to make it easier. In the next example, we will see $u$-substitution at work.

1. Use integration by substitution to find

   $$\int 2x\cos (x^2)dx$$

2. Using the substitution $u=x\ln x-x,$ find

   $$\int 3x^2\ln (x)(\ln (x)-1{)}^{2}dx$$