5'2 Integration by Substitution'

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5.2 Integration by Substitution.
The student will learn about
integration by substitution,
some substitution techniques, and
applications.
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Reversing the Chain Rule.
Recall the chain rule
This implies that
This means that in order to integrate u (x)
its derivative, u (x), must be present.
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General Integral Formulas
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Example
Note that the derivative of x 5 2 , (i.e. 5x
4 ), is present and the integral appears to be
in the chain rule form ? u (x) n u(x) dx,
with u(x) x 5 2 .
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Differential
If u f (x) is a differentiable function,
then 1. The differential dx of the
independent variable x is any arbitrary real
number.

• The differential du of the dependent variable
  u is defined as the product of u(x)
  and dx that is, as
• du u (x) dx

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Examples

• If u x 5 2 , then
• du u(x) dx 5x 4 dx

2. If u e 5x , then du u(x) dx
5e 5x dx
3. If u ln (3x -5), then du u(x) dx

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Integration by Substitution
(Substitution) Find ? (x2 1)5 (2x) dx
For our substitution let u x2 1,
2x, and du
2x dx
Then du/dx
and the integral becomes
? u5 du which is
u6/6 C
and reverse substitution yields
(x2 1)6/6 C.
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General Indefinite Integral Formulas.
? un du
Very Important
? e u du e u C
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Integration by Substitution.
Step 1. Select a substitution that appears to
simplify the integrand. In particular, try to
select u so that du is a factor of the integrand.

Step 2. Express the integrand entirely in terms
of u and du, completely eliminating the original
variable.
Step 3. Evaluate the new integral, if possible.
Step 4. Express the antiderivative found in step
3 in terms of the original variable. (Reverse the
substitution.)
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Example 2
? (x3 - 5)4 (3x2) dx
Step 1 Select u.
Let u x3 - 5
and then du
3x2 dx
Step 2 Express integral in terms of u.
? (x3 - 5)4 (3x2) dx
? u4 du
Step 3 Integrate.
? u4 du
u5/5 c
Step 4 Express the answer in terms of x.
u5/5 c
(x3 5) 5/5 c
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Example 3
? (x 2 5) 1/2 (2x ) dx
Step 1 Select u.
Let u x2 5
and then du
2x dx
Step 2 Express integral in terms of u.
? (x2 5) 1/2 (2x) dx
? u 1/2 du
Step 3 Integrate.
? u 1/2 du
u 3/2/(3/2) c 2/3 u 3/2 c
Step 4 Express the answer in terms of x.
2/3 u 3/2 c
2/3 (x2 5) 3/2 c
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Substitution Technique 1 Example 1
1.
? (x3 - 5)4 (x2) dx
Let u
x3 5 then du
3x2 dx
? (x3 - 5)4 (x2) dx
1/3 ? (x3 - 5)4 (3x2) dx
1/3 ? u4 du
1/3 u5/5
1/15 u5
Note we need a 3.
1/15 (x3 5)5 c
In this problem we had to insert a multiple 3 in
order to get things to work out. We did this by
also dividing by 3 elsewhere.
In this problem we had to insert a multiple 3 in
order to get things to work out. We did this by
also dividing by 3 elsewhere. Caution a
constant can be adjusted but a variable cannot.
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Substitution Technique 1 Example 2
2.
Let u
4x3 then du
12x2 dx
Note we need a 12.
1/12?eu du
1/12 eu c
In this problem we had to insert a multiple 12 in
order to get things to work out. We did this by
also dividing by 12 elsewhere.
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Substitution Technique 1 Example 3
3.
Let u
5 2x2 then du
- 4x dx
Note we need a - 4.
-1/4 u - 4 /-4
1/16 (5 2x2) - 4 c
In this problem we had to insert a multiple - 4
in order to get things to work out.
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Substitution Technique 2 Example 1
4.
Let u
x 6 then du
dx
?x (x 6)8 dx
and this is ok, but
? x u8 du
we need to get rid of the x.
However, since u x 6,
x
u 6, so the integral becomes
?(u 6) u8 du
? u9 6u8 du
u10/10 6 u9/9 c
1/10 (x 6)10 2/3 (x 6)9 c
In this problem we had to be somewhat creative
because of the extra x.
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Applications
The marginal price of a supply level of x bottles
of baby shampoo per week is given by
Find the price-supply equation if the distributor
of the shampoo is willing to supply 75 bottles a
week at a price of 1.60 per bottle.
To find p (x) we need the
? p (x) dx
Continued on next slide.
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Applications - continued
The marginal price of a supply level of x bottles
of baby shampoo per week is given by
Find the price-supply equation if the distributor
of the shampoo is willing to supply 75 bottles a
week at a price of 1.60 per bottle.
Let u
3x 25 and du
3 dx
Continued on next slide.
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Applications - continued
The marginal price of a supply level of x bottles
of baby shampoo per week is given by
Find the price-supply equation if the distributor
of the shampoo is willing to supply 75 bottles a
week at a price of 1.60 per bottle.
With u 3x 25,
becomes
Continued on next slide.
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Applications - continued
The marginal price of a supply level of x bottles
of baby shampoo per week is given by
Find the price-supply equation if the distributor
of the shampoo is willing to supply 75 bottles a
week at a price of 1.60 per bottle.
Now we need to find c using the fact
That 75 bottles sell for 1.60 per bottle.
and c
2
Continued on next slide.
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Applications - continued
The marginal price of a supply level of x bottles
of baby shampoo per week is given by
Find the price-supply equation if the distributor
of the shampoo is willing to supply 75 bottles a
week at a price of 1.60 per bottle.
So
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Summary.
1. ? un du
? e u du e u C
2.
Very Important
3.