WEBVTT

1
00:00:00.000 --> 00:00:04.122
Here is the problem: integrate x squared plus 1 from 1 to 2.

2
00:00:04.572 --> 00:00:11.097
Evaluate the definite integral of x squared plus 1 from x equals 1 to x equals 2.

3
00:00:11.897 --> 00:00:13.755
Start with the definite integral.

4
00:00:14.205 --> 00:00:21.647
Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

5
00:00:22.097 --> 00:00:26.416
Constant rule: the integral of k dx equals kx.

6
00:00:26.866 --> 00:00:29.246
Sum rule: integrate term by term.

7
00:00:30.046 --> 00:00:31.044
Antiderivative.

8
00:00:31.495 --> 00:00:34.049
Apply the fundamental theorem of calculus.

9
00:00:34.849 --> 00:00:40.909
So the answer is 10 over 3 approximately 3.3333333.

10
00:00:41.359 --> 00:00:50.368
A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

11
00:00:51.168 --> 00:00:56.184
Differentiated the antiderivative and compared it with the integrand, which checks out.

12
00:00:56.634 --> 00:01:04.448
Adaptive Simpson prime s rule gives 3.333333333, which checks out.