in this section, we will begin investigating some more advanced techniques for integration –...

In this section, we will begin investigating some more advanced techniques for integration – specifically, integration by parts.

Section 8.1 Integration By Parts

Idea

Integration by Parts is essentially a product rule for antidifferentiation.

It comes from the product rule from derivatives, as we will see in the proof of the theorem.

TheoremIntegration by Parts

If u and v are differentiable functions, then

1.

2.

Choosing u and dv

dv should be selected so that v can be found by antidifferentiating

Also, should be simpler to work with than the original integral.

!!! Do not forget to consider a variable substitution before using any more advanced technique.

Choosing u and dv

L I A T E

Choose u in this order:

L = logarithm

I = inverse trigonometry

A = algebraic functions

T = trigonometric functions

E = exponential functions

dv = rest of the original integrand

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Example 7

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Example 8

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Example 9

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