Indefinite Integrals explanations, examples, practice problems. Ready? Let's do this.
Indefinite Integrals Introduction
In this unit, we'll discuss techniques for finding integrals, both definite and indefinite. The first technique, integration by substitution, is a way of thinking backwards. Then we'll directly app...
Integration by Substitution: Indefinite Integrals
Integration by substitution is a way of undoing the chain rule. This is a once-in-a-lifetime opportunity to learn derivatives inside and out, forwards and backwards. Exciting, eh? Learning integrat...
Integration by Substitution: Definite Integrals
Be Careful: There are two ways to use substitution to evaluate definite integrals. When evaluating a definite integral, make sure you know which way you're using them. Way 1: First integrate the in...
Integration by Parts: Indefinite Integrals
We can think of integration by parts as a way to undo the product rule. While integration by substitution lets us find antiderivatives of functions that came from the chain rule, integration by par...
Integration by Parts: Definite Integrals
As with integration by substitution, there are two distinct ways to integrate definite integrals using integration by parts. As with integration by substitution, we have to be careful not to mix th...
Integration by Partial Fractions
Integration by partial fractions is a technique we can use to integrate rational functions when the degree of the numerator is less than the degree of the denominator. Here's the big picture:We sta...
Integrating Definite Integrals
More good news about integrating by partial fractions: there's only one way to integrate definite integrals. Find an antiderivative of the integrand. Use the Fundamental Theorem of Calc...
Choosing an Integration Method
We've been learning the different methods of integration in a very artificial environment. We know that if we're in the "Integration by Substitution" section, we use substitution. If we're in the "...
Improper Integrals
We'd like to introduce a couple of new words to help us talk about limits. If you're rusty on how limits work, we recommend reviewing them.When a limit exists and equals L, we say that limit conver...
Comparing Improper Integrals
Usually it's more important to know whether an improper integral converges than it is to know what it converges to. We can often figure out whether an improper integral converges or diverges by c...
In the Real World
We'll be honest: a lot of the mechanical integration methods you're learning here probably won't be that useful in the long run. Once you get out of school and into a real-life situation, you'll ge...