Quick u substitution integral(Online calculus course test homework solutions)

Question: Compute the integral using U substitution:



Solution:
At the onset, this doesn't look like a familiar derivative, so we want to make a substitution that allows us to perform a simple antiderivative. Accordingly, define U=2x+2. Then, by differentiation, we have that dU=2dx, or equivalently, dx=dU/2. So this transforms the integral as follows:



Notice how the lower and upper limits changed. Since u=2x+2, when x=19, u=40 and when x=22, u=46.

Now we have a simple integral that we now how to antidifferentiate: