Integration Techniques

The notes below provide an overview of common integration techniques. The notes that follow assume a strong background in trigonometry.

Trigonometric Integrals

Below are the basic trigonometric integrals:

\int \cos x ~dx = \sin x + C

\int \sin x ~dx = - \cos x + C

\int \sec^2 x ~dx = \tan x + C

\int \csc^2 x ~dx = -\cot x + C

\int \sec x \tan x ~dx = \sec x + C

\int \csc x \cot x ~dx = -\csc x + C

\int \tan x ~dx = \ln \abs{\sec x} + C

\int \cot x ~dx = \ln \abs{\sin x} + C

\int \sec x ~dx = \ln \abs{\sec x + \tan x} + C

\int \csc x ~dx = \ln \abs{\csc x - \cot x} + C

We now turn to slightly more complicated trigonometric integrals.

Dealing with Exponents

One of the most common forms of real-world trigonometric integrals is the following:

\int \sin^n (x) \cos^m(x) ~dx

When we encounter these integrals, the first question to ask is:

Are any of the exponents odd?

For example, suppose we had this integral:

\int \sin^n(x) \cos(x) ~dx

We already know what the integral of ${\cos x ~dx}$ is, so we make a substitution:

u = \sin(x)

Now, when we make this substitution, we must change everything in the expression that involves ${\sin(x).}$ This is so important to keep in mind. Substitution is an all-or-nothing game. Forget this fact, and we don't just lose — we're sent to the shadows. As such, we can't just get rid of ${\sin(x)}$ and call it a day. We must also get rid of the ${\cos(x) ~dx:}$

u = \sin(x) ~~~~ du = \cos(x) ~dx

Thus, we now have:

\int u^n ~du

This is easy to integrate:

\int u^n ~du = \dfrac{u^{n+1}}{n+1} + C

Substituting for ${u = \sin(x),}$ we get our answer:

\int \sin^n(x) \cos(x) ~dx = \dfrac{\sin^{n+1} x}{n+1} + C

In this example, we relied on the fact that ${\cos(x)~dx}$ has an odd power (in this case, 1), allowing us to use it as a differential.