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Examples

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...

Legrange (Prime) Notation

When we use the chain rule to take derivatives, there are some patterns that show up a lot. Some examples areWe can use these patterns to find derivatives.The general strategy for integration by su...

Leibniz (Fraction) Notation

To do integration by substitution using Leibniz notation, we think of the derivative function  as a fraction of infinitesimally small quantities du and dx. We change variables by manipulating...

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

You 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 pa...

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...

Badly Behaved Limits

Improper integrals with badly behaved limits are integrals where one or both of the limits is infinite. These integrals look likeIf only one limit of integration is infinite then the other limit of...

Badly Behaved Functions

Improper integrals with badly-behaved functions are deceptive. They look like normal definite integrals,but somewhere in the interval from a to b, possibly at one of the endpoints, there will be a...

Badly Behaved Everything

If you're given some random integral to integrate, you probably won't be told whether it's improper or not. It might be improper because of badly behaved limits, a badly behaved function, or both....

The p-Test

We often use integrals of the functions , for various values of p, to help determine whether other integrals converge or diverge.You already did the work to show this, so we'll just summarize the r...

Finite and Infinite Areas

Besides the p-test, there are a few basic principles that go into the rest of this section.If we have a convergent integral and we make its interval of integration smaller, the new integral will al...

Comparison with Formulas

We can figure out whether integrals converge or diverge by comparing them with other integrals whose convergence or divergence we already know. When we're looking at formulas and not at graphs, we...
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