Computing a definite integral using parts

When you compute a definite integral using integration by parts, there's nothing new as far as the integration goes. The only thing that's new is plugging in the limits of integration.

Suppose I want to find the area of the region under the curve y = x^2 sin x from x = 0 to x = pi. The curve and the region are shown below.

[Picture: y = x^2 sin x]

The area is given by the integral of x^2 sin x from x = 0 to x = pi.

[Picture: The integral for the area under y = x^2 sin x]

Compute the integral using a parts table:

[Animation: Parts table for the integral of x^2 sin x]

Now that I've finished the antiderivative, I can apply the limits and finish the computation:

[Picture: Plugging in the limits]

Introduction: Integration by parts
How do you set up an integration by parts table?
The integral of x^2 e^x
- How do you choose what to put in the table?
The integral of ln x
The integral of e^(2x) cos (3x)
A recursion formula for the integral of (sin x)^n
Computing a definite integral using integration by parts

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Last updated: June 13, 2005

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