Table of Contents

Indefinite Integrals of Trigonometric Functions
The indefinite integrals for $\sin x$ and $\cos x$ should be rather easy to remember as we are just going in the opposite cyclic pattern for differentiation. Recall that the cycle of $\sin x \to \cos x \to \sin x \to \cos x \to \sin x \to ...$ is a good memory device for remembering the derivatives of a function. For example, the derivative of $\sin x$ is $\cos x$ (by following the arrows). For integration, the same mechanic works just in the opposite direction, so the antiderivative of $\cos x$ is $\sin x$. This information is summarized in the following property:
Theorem 1: The following functions have the following indefinite integrals: a) If $f(x) = \sin x$, then $\int \sin x \: dx = \cos x + C$. b) If $f(x) = \cos x$, then $\int \cos x \: dx = \sin x + C$. c) If $f(x) = \tan x$, then $\int \tan x \: dx = \ln \mid \sec x \mid + C$. 
The indefinite integral for $\tan x$ is a little more complicated, and we cannot verify this integral yet. Let's look at some examples.
Example 1
Evaluate the following integral: $\int 5 \cos x \: dx$.
From the Properties of Indefinite Integrals, and from the rules above, we can simplify this:
(1)Example 2
Evaluate the following integral: $\int 4t^3 + \sin t \: dt$.
Once again, we will apply what we know:
(2)Example 3
Evaluate the following integral: $\int \frac{\sin x}{\cos x} + \tan x  x^\pi \: dx$.
Recall that $\frac{\sin x}{\cos x} = \tan x$, so it follows that:
(3)Now we can integrate:
(4)