# Using Variable Substitution to Evaluate Definite Integrals

When using variable substitution to evaluate a definite integral, you can save yourself some trouble at the end of the problem. Specifically, you can leave the solution in terms of *u *by changing the limits of integration.

For example, suppose that you’re evaluating the following definite integral:

Notice that this example gives the limits of integration as *x* = 0 and *x* = 1. This is just a notational change to remind you that the limits of integration are values of *x.* This fact becomes important later in the problem.

You can evaluate this equation simply by using variable substitution:

If this were an indefinite integral, you’d be ready to integrate. But because this is a definite integral, you still need to express the limits of integration in terms of *u* rather than *x.* Do this by substituting values 0 and 1 for *x* in the substitution equation *u* = *x*^{2} + 1:

*u* = 1^{2} + 1 = 2

*u* = 0^{2} + 1 = 1

Now use these values of *u* as your new limits of integration:

At this point, you’re ready to integrate:

Because you changed the limits of integration, you can now find the answer without switching the variable back to *x:*