Integrating Sin² X

The solution to the integral of sin^2(x) requires you to recall principles of both trigonometry and calculus. Don’t conclude that since the integral of sin(x) equals -cos(x), the integral of sin^2(x) should equal -cos^2(x); in fact, the answer does not contain a cosine at all. You cannot directly integrate sin^2(x). Use trigonometric identities and calculus substitution rules to solve the problem.

Step 1

Use the half angle formula, sin^2(x) = 1/2*(1 – cos(2x)) and substitute into the integral so it becomes 1/2 times the integral of (1 – cos(2x)) dx.

Step 2

Set u = 2x and du = 2dx to perform u substitution on the integral. Since dx = du/2, the result is 1/4 times the integral of (1 – cos(u)) du.

Step 3

Integrate the equation. Since the integral of 1du is u, and the integral of cos(u) du is sin(u), the result is 1/4*(u – sin(u)) + c.

Step 4

Substitute u back into the equation to get 1/4*(2x – sin(2x)) + c. Simplify to get x/2 – (sin(x))/4 + c.

TL;DR (Too Long; Didn’t Read)

For a definite integral, eliminate the constant in the answer and evaluate the answer over the interval specified in the problem. If the interval is 0 to 1, for example, evaluate [1/2 – sin(1)/4] – [0/2 – sin(0)/4)].