Integrating Rational Functions Integrating Rational Functions

Integrating Rational Functions

1. IntegratingIntegrating
Rational FunctionsRational Functions
TS: Making decisions after reflection and reviewTS: Making decisions after reflection and review

2. ObjectiveObjective
To evaluate the integrals of exponential To evaluate the integrals of exponential
and rational functions.and rational functions.

3. Remember…Remember…
What is the derivative of ln(u)?What is the derivative of ln(u)?
1
du
u
1
ln| |du u C
u
 
1
du
u

So what is theSo what is the again?again?

4. Exponential FunctionsExponential Functions

3
2
2
Evaluate
1
x
dx
x

2
Let 1u x 
2 du xdx
3
2

x
u
dx
3
u du


2
2
u
C


 
2
1
2u
C 
3
2 u xdx

 

2
2
1
2 1x
C

 

5. Exponential FunctionsExponential Functions
2
2
Evaluate
1
x
dx
x

2
Let 1u x 
2 du xdx
2

x
u
dx
1

u
du
lnu C 
2
ln 1x C  
1
2
u
xdx 

2
ln 1x C  

6. Exponential FunctionsExponential Functions
Evaluate
1
x
x
e
dx
e

Let 1
x
u e 

x
du e dx

x
e
u
dx
1

u
du
lnu C 
ln 1
x
e C  
1

x
u
e dx 
ln 1
x
e C  

7. ConclusionConclusion
Integration by substitution is a technique for Integration by substitution is a technique for
finding the antiderivative of a composite finding the antiderivative of a composite
function.function.
Experiment with different choices for Experiment with different choices for uu when when
using integration by substitution.using integration by substitution.
A good choice is one whose derivative is A good choice is one whose derivative is
expressed elsewhere in the integrand.expressed elsewhere in the integrand.