Chapter 7.2, Problem 49E

To determine

To determine the integral ${\displaystyle \int 2{\sin }^2(x)dx}$ by substitution.

Answer to Problem 49E

The value of the integral ${\displaystyle \int 2{\sin }^2(x)dx}$ is $x-\frac{\sin (2x)}{2}+C$

Explanation of Solution

Given information: ${\displaystyle \int 2{\sin }^2(x)dx}$

Formula used: ${\displaystyle \int \cos (x)dx=\sin (x)+C}$

Calculation :

The given integral becomes as:

  ${\displaystyle \int 2\left(\frac{1-\cos (2x)}{2}\right)dx}={\displaystyle \int 1-\cos (2x)dx}$   $\left[\text{Since},{\sin }^2(x)=\frac{1-\cos (2x)}{2}\right]$

  $=x-\frac{\sin (2x)}{2}+C$

Hence, ${\displaystyle \int 2{\sin }^2(x)dx}=x-\frac{\sin (2x)}{2}+C$