Calculus:
Integral: Trigonometric Functions

1.  
\(\int \tan^{2}x \, dx\)
  Solution
2.  
\(\int x+\sec^{3}x \, dx\)
  Solution
3.  
\(\int \tan^{3}x \, dx\)
  Solution
4.  
\(\int \cot^{4}x+{x}^{3} \, dx\)
  Solution
5.  
\(\int \csc^{3}x \, dx\)
  Solution
6.  
\(\int \cos^{2}x+{x}^{3} \, dx\)
  Solution
7.  
\(\int \sin^{2}x \, dx\)
  Solution
8.  
\(\int \sec{x}+{e}^{x} \, dx\)
  Solution
9.  
\(\int \sin^{3}x \, dx\)
  Solution
10.  
\(\int \cos^{3}x \, dx\)
  Solution

Integrals with Trigonometric Functions: Introduction

Trigonometry covers a subset of mathematical functions and concepts that many students (and parents) find daunting — how do the six trigonometric functions work together? How can they be manipulated effectively, and how are they used across derivative and integral equations?

Trigonometry Basics

Let us not forget the basics: Trigonometry centers around the degree and radian measurements of a right triangle — thus, the “tri” in "trigonometry". There are six basic trigonometric functions, which are found by dividing one side a triangle by another: In a right triangle, the “long” side is the hypotenuse; the side closest to the angle being measured is the adjacent; and the remaining side is the opposite. Dividing each side by the other, in turn, gives us the six basic trigonometric functions:
\(\cos{x}\)
,
\(\sin{x}\)
,
\(\tan{x}\)
,
\(\sec{x}\)
,
\(\cot{x}\)
and
\(\csc{x}\)
.

Solving Integrals with Trigonometric Functions

What happens when you encounter integrals that involve these trigonometric functions? Generally speaking, there are a few main strategies, and often multiple strategies are used together to find the integral. They are:
1. Trigonmetric Identities
2. Integration by Substitution
3. Integration by Reduction Formula
Let’s try an example. Consider this integral:
\(\int \tan^{2}x \, dx\)
How do we integrate this? Let's first ask ourselves: is there a trignometric identity that can transform this integral into something easier to solve? The Pythagorean Identities might come to mind. The three Pythagorean Identities are:
1.
\(\sin^{2}x+\cos^{2}x=1\)

2.
\(\tan^{2}x+1=\sec^{2}x\)

3.
\(\cot^{2}x+1=\csc^{2}x\)
Since our integral contains
\(\tan^{2}x\)
, let's see if we can use the second identity above. Rearranging the terms gives us this equation:
\(\tan^{2}x=\sec^{2}x-1\)
.
This lets us rewrite the integral as:
\(\int \sec^{2}x-1 \, dx\)
Now, we can use the sum rule, which states:
\(\int f(x)+g(x) \, dx=\int f(x) \, dx+\int g(x) \, dx\)
. This allows us to rewrite the integral as:
\(\int \sec^{2}x \, dx-\int 1 \, dx\)
Since we know the derivative of
\(\tan{x}\)
is
\(\sec^{2}x\)
, it follows that the integral of
\(\sec^{2}x\)
must be
\(\tan{x}\)
. This gives us
\(\tan{x}-\int 1 \, dx\)
. Finally,
\(\int 1 \, dx\)
is simply
\(x\)
by the Power Rule. Together, our final answer is:
\(\int \tan^{2}x \, dx=\tan{x}-x+C\)

What's Next

You can try more practice problems at the top of this page to help you get more familiar with solving integral with trigonometric functions. Want even more help? Sign up for Cymath Plus today. Download the Cymath app for iOS and Android to get step-by-step assistance anytime, anywhere.
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