Unit 7A

Integrals Part 1

Lesson 6: Integration by Parts

Integration by Parts

Integration by parts is a general technique of integration. Based on the derivative of a product, this technique is used to express an integral that is difficult to evaluate in terms of a second integral that is easier to evaluate.

The development of integration by parts is as follows.

Assume «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» are differentiable functions.

Recall the product rule.


Integrate both sides.


Simplify.


Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» and let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math».

Substitute «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mo»=«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi»g«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»+«/mo»«mo»§#8747;«/mo»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/math».

     
Rearranging the equation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«mi»v«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mi»u«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»v«/mi»«mo»+«/mo»«mo»§#8747;«/mo»«mi»v«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»u«/mi»«/math» results in the integration by parts formula.


The examples that follow will show that with careful selection of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»u«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«/math» in the original problem, the integral «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»v«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»u«/mi»«/math» will be easier to evaluate than the original integral «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»u«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»v«/mi»«/math».