WEBVTT

1
00:00:00.000 --> 00:00:03.854
Here is the problem: the integral of sin(x) squared dx.

2
00:00:04.304 --> 00:00:09.772
Find the indefinite integral (antiderivative) of sin(x) squared with respect to x.

3
00:00:10.572 --> 00:00:12.058
Start with the integral.

4
00:00:12.508 --> 00:00:18.081
Power-reducing identity: sin squared u equals (1 minus cos 2u)/2.

5
00:00:18.531 --> 00:00:21.468
Simplify and add the constant of integration.

6
00:00:22.268 --> 00:00:27.620
So the answer is x / 2 minus sin(2x) / 4 plus C.

7
00:00:28.070 --> 00:00:39.146
An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

8
00:00:39.946 --> 00:00:45.844
Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.