Integration by Parts

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Description:

Students should be able to evaluate definite and indefinite integrals using integration by parts.

This packet consists of five videos that introduce the concepts of integration by parts, examine some techniques to be used when integrating by parts, and walk through several examples.

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Tutorial

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.

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Integration by Parts - LIATE

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

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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.

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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.

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Integration by Parts - Examples

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

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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: $pi over 4 plus 1 half ln left parenthesis 2 right parenthesis$