Integration with trigonometric identities

In a nutshell

Many integrals can be simplified by using any number of trigonometric identities.

List of trigonometric identities

Here is a list of all the trigonometric identities you need to know.

These can be used to make certain integrals much simpler.

Example 1

Compute the following integrals:

i) $$\int \cos^2(x)\,dx$$​

ii) $$\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx$$​

Part i):

Use the double angle formula to express $$\cos^2(x)$$ in terms of $$\cos(2x)$$:

​$$\begin{aligned} \cos(2x)&\equiv 2\cos^2(x)-1\\ \Rightarrow \cos^2(x)&\equiv \dfrac{\cos(2x)+1}{2} \end{aligned}$$​​

This is easier to integrate:

$$\begin{aligned} \int \cos^2(x) \, dx&\equiv \int \left(\dfrac{\cos(2x)+1}{2}\right) \, dx\\&= \dfrac12\int \cos(2x)\, dx + \int\dfrac12 \, dx\\&=\dfrac12 \left(\dfrac12 \sin(2x)\right) + \dfrac12 x+C\\&=\dfrac14 \sin(2x) + \dfrac12x+C \end{aligned}$$​​

​$$\int \cos^2(x) =\underline{\dfrac14 \sin(2x) + \dfrac12x+C}$$​​

Part ii):

Simplify the integrand by writing everything as one fraction:

$$\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx \equiv \int\dfrac{1}{\cosec(2x)}\left( \dfrac{\cos^2(x)+\sin^2(x)}{\sin(x)\cos(x)}\right)\,dx$$​​

Use the following identities to simplify this:

$$\cos^2(x)+\sin^2(x) \equiv 1\\ \sin(2x)\equiv 2\sin(x)\cos(x) \Rightarrow \sin(x)\cos(x) \equiv \dfrac12 \sin(2x)\\ \cosec(x)\equiv \dfrac{1}{\sin(x)} \Rightarrow \cosec(2x) \equiv \dfrac{1}{\sin(2x)}$$​​

The integral becomes:

$$\begin{aligned} \int\dfrac{1}{\cosec(2x)}\left( \dfrac{\cos^2(x)+\sin^2(x)}{\sin(x)\cos(x)}\right)\,dx &=\int \dfrac{1}{\frac{1}{\sin(2x)}} \left(\dfrac{1}{\frac12 \sin(2x)}\right) \, dx\\&=\int \sin(2x)\times \dfrac{2}{\sin(2x)}\, dx\\ &=\int 2\, dx\\&=2x+C \end{aligned}$$​​

​$$\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx=\underline{2x+C}$$​​