L2 Basic Integration Properties and Techniques - Part 1
Completion requirements
Unit 7A
Integrals Part 1
Lesson 2: Basic Integration Properties and Techniques
In Lesson 1, several integration properties were introduced. Some of those techniques will now be formally stated.
In the following properties, «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» represent differentiable functions, «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»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» are their respective derivatives, and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» are constants.
These rules will be applied to polynomials and simple algebraic functions.
In the following properties, «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» represent differentiable functions, «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»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» are their respective derivatives, and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» are constants.
Property 1:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»)«/mo»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»F«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»C«/mi»«/math»
Property 2:
The constant «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» may be moved across the integral sign.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»a«/mi»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mi»a«/mi»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«/math»
Property 3:
The integral of a sum or difference is the sum or difference of the integrals.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#177;«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»§#177;«/mo»«mo»§#8747;«/mo»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»)«/mo»«mi»d«/mi»«mi»x«/mi»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#177;«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8747;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»§#177;«/mo»«mo»§#8747;«/mo»«mi»g«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»)«/mo»«mi»d«/mi»«mi»x«/mi»«/math»
Property 4:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8747;«/mo»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/msup»«mrow»«mi»n«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mi»where«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»n«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/math»
These rules will be applied to polynomials and simple algebraic functions.