Question

How do you find the antiderivative of ${{\left( \cos \left( 2x \right) \right)}^{2}}$?

Answer 1

In this problem we need to calculate the antiderivative of the given function which is nothing but the integration of the given function. We can observe that the given equation contains the trigonometric function ${{\cos }^{2}}2x$. We will write this trigonometric function as $\dfrac{\cos 4x+1}{2}$. Now we will apply the integration to the above value. Here we will apply the integration to individual terms and use the integration formulas and simplify the equation to get the required result.

Complete step by step answer:
Given function, ${{\left( \cos \left( 2x \right) \right)}^{2}}$.
In trigonometry we have the formula $\cos 2x=2{{\cos }^{2}}x-1$. From this formula we can write the value of ${{\cos }^{2}}2x$ as
$\Rightarrow {{\cos }^{2}}2x=\dfrac{\cos 4x+1}{2}$
Integrating both sides of the above equation, then we will get
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\int{\dfrac{\cos 4x+1}{2}dx}$
Applying the integration to all the terms individually in the above equation, then we will have
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\dfrac{1}{2}\int{\cos 4xdx}+\dfrac{1}{2}\int{dx}$
We have the integration formula $\int{\cos axdx}=\dfrac{\sin ax}{a}+C$. Applying this formula in the above equation, then we will get
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\dfrac{1}{2}\left[ \dfrac{\sin 4x}{4} \right]+\dfrac{1}{2}\int{dx}+C$
Simplifying the above equation, then we will have
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\dfrac{\sin 4x}{8}+\dfrac{1}{2}\int{dx}+C$
Now we have the integration formula $\int{dx}=x+C$. Applying this formula in the above equation, then we will get
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\dfrac{\sin 4x}{8}+\dfrac{1}{2}\left( x \right)+C$
Simplifying the above equation, then we will have
$\Rightarrow \int{{{\cos }^{2}}2xdx}=\dfrac{\sin 4x}{8}+\dfrac{x}{2}+C$

Note: For this problem we can also use the substitution method to get the required result. You can consider the substitution $u=2x$, the differentiation of above value $du=2dx$. We will use both the above values and integration formulas to get the integration value of the given equation.