WEBVTT

1
00:00:00.000 --> 00:00:04.261
Here is the problem: integrate sin(x) to the power 2cos(x).

2
00:00:04.711 --> 00:00:10.829
Find the indefinite integral (antiderivative) of sin(x) squared cos(x) with respect to x.

3
00:00:11.629 --> 00:00:13.115
Start with the integral.

4
00:00:13.565 --> 00:00:18.232
Substitute u equals sin(x), so du equals (cos(x)) dx.

5
00:00:18.682 --> 00:00:26.031
Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

6
00:00:26.481 --> 00:00:29.256
Substitute back u equals sin(x).

7
00:00:30.056 --> 00:00:32.993
Simplify and add the constant of integration.

8
00:00:33.793 --> 00:00:37.648
So the answer is sin(x) cubed/3 plus C.

9
00:00:38.097 --> 00:00:48.720
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.

10
00:00:49.520 --> 00:00:55.360
Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.