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Lecture

MAT136H1 Lecture Notes - Partial Fraction Decomposition, Function Type


Department
Mathematics
Course Code
MAT136H1
Professor
all

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7.4 Integration Techniques
Partial Fractions for Rational Functions
Question #5 (Medium): Rationalizing Substitutions
Strategy
Non-rational functions refer to those containing root functions in either the numerator or denominator,
but also those that include trigonometric or exponential functions. Either way, a simple variable is used
to substitute as the building block to represent the function type. Then factor the denominator, split the
rational function, then take integral. Remember to re-substitute back in the original expression.
Sample Question
By substitution express the integral as a rational function and then evaluate the integral.


Solution
First factor the function in the denominator, so that: .
Notice the rational function is in the form of chain rule. So let  and . Then:


. Apply partial fractions: 



. When , then , so
.
 means 
, so
.
Then:



 




Substitute back in the original expression:












Therefore, the integral is evaluated as: 



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