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Integration by Parts

Author: Richard Enderton

Prerequisites

Student should be familiar with evaluating definite and indefinite integrals using the Power Rule, Basic Formulas, and u-Substitution. Students should also be familiar with differentiation, particularly the Product Rule.

Integration by Parts - the Formula

This video develops the formula and technique for integration by parts.

Integration by Parts - LIATE

This video looks at the acronym LIATE and how it can help making good choices when applying integration by parts.

Integration by Parts - the Table Method

This video looks at a quick way of evaluating certain types of integrals that involve repeated application of integration by parts.

Integration by Parts - the Grid

This video looks at a techniques used by many to help keep the different part straight when applying integration by parts.

Integration by Parts - Examples

This video walks through five more examples of integration by parts, including a couple of the more difficult situations.

Problem Set

Evaluate the indefinite/definite integrals.

NOTE that not all require integration by parts.

1. $integral x e to the power of 2 x end exponent d x$ Ans: $1 half x e to the power of 2 x end exponent minus 1 fourth e to the power of 2 x end exponent plus C$

2. $integral x to the power of 2 space end exponent ln x space d x$ Ans: $x cubed over 3 ln x minus 1 over 9 x cubed plus C$

3. $integral x squared space sin left parenthesis x cubed right parenthesis space d x$ Ans: $negative 1 third cos left parenthesis x cubed right parenthesis plus C$

4. $integral subscript 0 superscript pi divided by 4 end superscript x space cos x space d x$ Ans: $fraction numerator pi over denominator 4 square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction minus 1$ Ans:

5. $integral a r c sin x space d x$ Ans: $x a r c sin left parenthesis x right parenthesis plus square root of 1 minus x squared end root plus C$

6. $integral x to the power of 4 space e to the power of 1 half x end exponent space d x$

Ans: $2 x to the power of 4 e to the power of 1 half x end exponent minus 16 x cubed e to the power of 1 half x end exponent plus 96 x squared e to the power of 1 half x end exponent minus 384 x e to the power of 1 half x end exponent plus 768 e to the power of 1 half x end exponent plus C$

7. $integral e to the power of x space cos x space d x$ Ans: $fraction numerator e to the power of x cos x plus e to the power of x sin x over denominator 2 end fraction plus C$

8. $integral subscript pi divided by 4 end subscript superscript pi divided by 2 end superscript x space c s c squared x space d x$ Ans: $pi over 4 plus 1 half ln left parenthesis 2 right parenthesis$

Additional Resources

Paul' Online Notes

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx

A set of worked out examples

http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/intbypartsdirectory/IntByParts.html

More worked out examples

http://www.intmath.com/methods-integration/7-integration-by-parts.php

Wolfram Alpha