Understanding What Makes a Function Integrable
Substituting with Expressions of the Form f(x) Multiplied by h(g(x))
Use a Shortcut for Integrating Compositions of Functions

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:

image0.png

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:

image1.png

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 = x2 + 1:

u = 12 + 1 = 2

u = 02 + 1 = 1

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

image2.png

At this point, you’re ready to integrate:

image3.png

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

image4.png
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Knowing When to Avoid Trigonometry Substitution
How to Integrate a Function Multiplied by a Set of Nested Functions
Integrate a Function Using the Sine Case
Finding the Integral of a Product of Two Functions
Integrate a Function Using the Secant Case
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