integration by substitution antidifferentiation of a composite function
TRANSCRIPT
Integration by Substitution
Antidifferentiation of a Composite Function
Antidifferentiation of a Composite Function• Let g be a function whose range is an
interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then
'( ) ( ( )f g x g x dx F g x C
Exploration
• Recognizing Patterns. Discover the rule using the exploration
Recognizing the f’(g(x)) g’(x) Pattern
• Evaluate
32 1 2x x dx
3
24
'( ) 2 ( ( ))
( 1( )
4 4
g x x and f g x g x
xg xC C
Recognizing the f’(g(x)) g’(x) Pattern
• Evaluate
2
1 1cos d
2
2
1 1( ) '( )
1 1 1( ( )) cos cos
1sin
Letting g x and g x
f g x d
C
Multiplying and Dividing by a Constant
• Evaluate
324 3x x dx
32
32
4 42 2
Rewrite the integral as
4 3
14 3 8
8
4 3 4 31
8 4 32
x x dx
x x dx
x xC C
Evaluating an Integral
22t tdt
t
We presently have no division rule for integrals. As a matter of fact there will never be a rule that involves division.
Suggestions?
Change of Variables
• We are allowed to multiply any integral by a constant, but we can never multiply by a variable. What can we do if there is an extra variable in the given equation?
Integration by Substitution
• Evaluate
• There is an extra x in this integrand. In order to solve this equation, we will let
• u = 2x – 1.
2 1x x dx
Integration by Substitution
• Now solve this equation for x
12
1 Next find the derivative of this function
2
Then substitute the variables into the given integral21
Now symplify2 2
ux
dudx
u duu
Integration by Substitution
12
3 12 2
5 32 2 5 3
2 2
11
41
4
1 1 15 34 10 62 2
Now substitute (2 1) back in for .
u u du
u u du
u uC u u C
x u
Integration by Substitution
• This is a correct answer, but it can be simplified a little further. If the problem is not a multiple choice question on the test or quiz, this answer is completed. If it is a multiple choice question, it will be given as
5 3
2 21 1
2 1 2 110 6
x x C
Integration by Substitution
32
3 32 2
12 1 3 2 1 5
301 1
2 1 6 3 5 2 1 6 830 30
x x
x x x x
Integrating Trig Functions
• Remember that you need the derivative of the trig function and the function.
2
Evaluate
sin 3 cos3x x dx
Guidelines for Making a Change of Variable
1. Choose a substitution u = g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power.
2. Compute du = g’(x) dx.3. Rewrite the integral in terms of the variable u.4. Evaluate the resulting integral in terms of u.5. Replace u by g(x) to obtain an antiderivative in
terms of x.6. Check your answer by differentiating.
General Power Rule for Integration
• If g is a differentiable function of x, then
1
1( )
( ) '( ) , 11
Equivalently, if ( ) then
11
n
nn
n
g xg x g x dx C n
nu g x
uu du C n
n
Definite Integrals
3 2 3
0Evaluate ( 1)x x dx
3 32
0
3 34 4 42 2 2 2
00
11 2
2
1 1 3 1 0 11
2 4 8 8 8
10,000 1 9999
8 8 8
x x dx
x x
Integration of Even and Odd Functions
• Let f be integrable on the closed interval, [a, –a ],
• 1. If f is an even function, then
• 2. If f is an odd function, then
0( ) 2 ( )
a a
af x dx f x dx
( ) 0.a
af x dx