Integration techniques

Contents
We know that a definite integral can represent area and we've seen how this is connected to the idea of an antiderivative through the Fundamental Theorem of Calculus. Unfortunately, integrals aren't always easy to compute. Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!

Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.

u-substitution

U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverse chain rule. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

Reverse chain rule

The reverse chain rule is another, faster way to think about u-substitution.

Integration using trigonometric identities

You will occasionally encounter integrals in life that involve products and exponents of trig functions. In this tutorial, you will see examples of using trigonometric identities to get these types of integrals into a form that you can actually integrate.

Trigonometric substitution

Another super useful technique for computing integrals involves replacing variables with trigonometric functions. This can make things seem a little more complicated at first, but with the help of trigonometric identities, this technique makes certain integrals solvable.

Division and partial fraction expansion

When you're trying to integrate a rational expression, the techniques of partial fraction expansion and algebraic long division can be *very* useful.