WEBVTT

1
00:00:00.000 --> 00:00:05.039
Here is the problem: the integral of x/(x squared plus 4) dx.

2
00:00:05.489 --> 00:00:12.397
Find the indefinite integral (antiderivative) of x / (x squared plus 4) with respect to x.

3
00:00:13.197 --> 00:00:14.683
Start with the integral.

4
00:00:15.133 --> 00:00:24.781
Logarithm rule: the numerator is a constant multiple of the denominator prime s derivative, the integral of u prime /u dx equals ln|u|.

5
00:00:25.231 --> 00:00:28.238
Simplify and add the constant of integration.

6
00:00:29.038 --> 00:00:33.694
So the answer is ln(abs(x squared plus 4)) / 2 plus C.

7
00:00:34.144 --> 00:00:44.918
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
00:00:45.718 --> 00:00:51.744
Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.