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Chapter Contents

• Methods of Integration
• 1. Integration: The General Power Formula
• 2. Integration: The Basic Logarithmic Form
• 3. Integration: The Exponential Form
• 4. Integration: The Basic Trigonometric Forms
• 5. Integration: Other Trigonometric Forms
• 6. Integration: Inverse Trigonometric Forms
• 7. Integration by Parts
• 8. Integration by Trigonometric Substitution
• 9. Integration by Use of Tables
• Table of Common Integrals
• 10. Integration by Reduction Formulae
• 11. Integration by Partial Fractions

• Comments, Questions?

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From the math blog...

7. Integration by Parts

by M. Bourne

If u and v are functions of x, the product rule for differentiation that we met earlier gives us:

Rearranging, we have:

Integrating throughout, with respect to x, we obtain the formula for integration by parts:

This formula allows us to turn a complicated integral into more simple ones. We must make sure we choose u and dv carefully.

Function u is chosen so that is simpler than u.

 

Priorities for Choosing u

1. Let u = ln x

2. Let u = xⁿ

3. Let u = e^nx

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Example 1:

Solution:

We could let u = x or u = sin 2x. In general, we choose the one that allows to be of a simpler form than u.

So for this example, we choose u = x and dv = sin 2x dx.

u = x	dv = sin 2x dx
du = dx

Substituting into the integration by parts formula, we get:

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Example 2:

Answer

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Example 3:

Answer

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Example 4:

Answer

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Example 5:

Answer

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Example 6:

Answer

Alternate Method for Integration by Parts

Here's an alternative method for problems that can be done using Integration by Parts. You may find easier to follow.

Tanzalin Method