# Integrating Trigonometric Functions by Double Angle Formula

## Integrating Trigonometric Functions by Double Angle Formula

Integrating Trigonometric Functions can be done by Double Angle Formula reducing the power of trigonometric functions.

\begin{aligned} \displaystyle \cos{2A} &= 2\cos^2{A}-1 \\ &= 1-2\sin^2{A} \\ &= \cos^2{A}-\sin^2{A} \end{aligned}

## Practice Questions

### Question 1

Find $\displaystyle \int{\cos^2{x}}dx$.

\begin{aligned} \displaystyle 2\cos^2{x}-1 &= \cos{2x} \\ 2\cos^2{x} &= \cos{2x} + 1 \\ \cos^2{x} &= \frac{1}{2}(\cos{2x}+1) \\ \int{\cos^2{x}}dx &= \frac{1}{2} \int{(\cos{2x}+1)}dx \\ &= \frac{1}{2}\Big(\frac{1}{2}\sin{2x}+x\Big) + C \\ &= \frac{1}{4} \sin{2x} + \frac{1}{2}x + C \end{aligned}

### Question 2

Find $\displaystyle \int{\sin^2{x}}dx$.

\begin{aligned} \displaystyle \cos{2x} &= 1-2\sin^2{x} \\ 2\sin^2{x} &= 1-\cos{2x} \\ \sin^2{x} &= \frac{1}{2}(1-\cos{2x}) \\ \int{\sin^2{x}}dx &= \frac{1}{2} \int{(1-\cos{2x})}dx \\ &= \frac{1}{2}\Big(x-\frac{1}{2}\sin{2x}\Big) + C \\ &= \frac{1}{2}x-\frac{1}{4}\sin{2x} + C \end{aligned}

### Question 3

Find $\displaystyle \int{\sin^4{x}}dx$.

\begin{aligned} \displaystyle \sin^4{x} &= (\sin^2{x})^2 \\ &= \frac{1}{4}(1-\cos{2x})^2 \\ &= \frac{1}{4}(1-2\cos{2x}+\cos^2{2x}) \\ &= \frac{1}{4}\Big[1-2\cos{2x}+\frac{1}{2}(1+\cos{4x})\Big] \\ &= \frac{1}{4}\Big[\frac{3}{2}-2\cos{2x}+\frac{1}{2}\cos{4x}\Big] \\ &= \frac{3}{8}-\frac{1}{2}\cos{2x}+\frac{1}{8}\cos{4x} \\ \int{\sin^4{x}}dx &= \int{\Big(\frac{3}{8}-\frac{1}{2}\cos{2x}+\frac{1}{8}\cos{4x}\Big)}dx \\ &= \frac{3}{8}x-\frac{1}{4}\sin{2x} + \frac{1}{32}\sin{4x} + C \end{aligned}

### Question 4

Find $\displaystyle \int{(\cos^2x-\sin^2x)}dx$.

\begin{aligned} \displaystyle \int{(\cos^2x-\sin^2x)}dx &= \int{\cos{2x}}dx \\ &= \frac{1}{2}\sin{2x} + C \end{aligned}

âœ“ Discover more enlightening videos by visiting our YouTube channel!

## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## The Best Practices for Using Two-Way Tables in Probability

Welcome to a comprehensive guide on mastering probability through the lens of two-way tables. If you’ve ever found probability challenging, fear not. We’ll break it…

## Binomial Expansions

The sum $a+b$ is called a binomial as it contains two terms.Any expression of the form $(a+b)^n$ is called a power of a binomial. All…

## High School Math for Life: Making Sense of Earnings

Salary Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis,…

## Quick Guide to Max Range in Inclined Plane Motion

Projectile motion on an inclined plane is a fascinating variation of the classic projectile motion problem. In this scenario, the trajectory of the projectile is…