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Chapter 5:
Integrals
Section 5.5
The Substitution Rule
(u-substitution)
Sec. 5.5: The Substitution Rule
• We know how to find the derivative of any
combination of functions
• Sum rule
• Difference rule
• Constant multiple rule
• Product rule
• Quotient rule
• Chain rule
Sec. 5.5: The Substitution Rule
• We would like to know how to find the
antiderivative of any combination of functions, but
only a few rules exist
• Sum rule
• Difference rule
• Constant multiple rule
• Product rule (no such rule)
• Quotient rule (no such rule)
• Chain rule (no such rule)
• This makes antiderivatives more difficult. We can’t
find the antiderivative of just any function
Sec. 5.5: The Substitution Rule
• What we’re going to do is learn integration
techniques: develop rules so that if you are in
certain very specific situations, you will be able to
find the antiderivative
• The substitution rule (today)
• Integration by parts (Calc. II)
• Integration by partial fractions (Calc. II)
• Trig. substitution (Calc. II)
Sec. 5.5: The Substitution Rule
U-Substitution:
• The idea is to reverse the chain rule
• Remember that whenever you know the derivative
of a function, you know an antiderivative of another
function
Ex: Since 𝑠𝑖𝑛 𝑥2 ′ = 2𝑥 𝑐𝑜𝑠 𝑥2 , we have …
2𝑥 𝑐𝑜𝑠 𝑥2 𝑑𝑥 = 𝑠𝑖𝑛 𝑥2 + 𝐶
Sec. 5.5: The Substitution Rule
When to use u-substitution (hints):
• When it works!
• If the derivative of part of the function you’re
integrating is another part of the function you’re
integrating
• Make u the part of the integrand (thing you’re
integrating) whose derivative is also a part of the
thing you’re integrating
• The derivative of part of the function needs to be
multiplied by that part
• The number in front of the derivative doesn’t
matter
Sec. 5.5: The Substitution Rule
What performing a u-substitution:
• Rewrite the integral so that it only has u’s in it
• Make sure the new integral is simpler than the
original integral
Sec. 5.5: The Substitution Rule
U-Substitution To Find An Indefinite Integral:
• Let u be the appropriate part of the function
• Find du, then solve for dx
• Rewrite the integral in terms of u (at the end of
this step only u should appear in the integral)
• The new u integral should be simpler than the
original integral
• Integrate with respect to u
• At the end, replace u with what you said u equals
at the beginning of the problem because the final
answer should have x’s in it, not u’s.
• Don’t forget the +C
Sec. 5.5: The Substitution Rule
U-Substitution To Find A Definite Integral:
Start off as if you are calculating an indefinite integral.
Then there are 2 ways to go
1) x-worldu-worldx-worldplug in the numbers
2) x-worldu-world
change the limits of integration to u numbers and
never go back to x world
Sec. 5.5: The Substitution Rule
Look at some problems
2𝑥 𝑠𝑖𝑛 𝑥2 𝑑𝑥
𝑢 = 𝑥2
Sec. 5.5: The Substitution Rule
Look at some problems
cos 𝑥 𝑒sin 𝑥 𝑑𝑥
𝑢 = sin 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
𝑠𝑖𝑛−1𝑥
1 − 𝑥2 𝑑𝑥
𝑢 = 𝑠𝑖𝑛−1𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
(ln 𝑥)2
𝑥 𝑑𝑥
𝑢 = ln 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
−3𝑥2 2𝑥3 + 7 𝑑𝑥
𝑢 = 2𝑥3 + 7
Sec. 5.5: The Substitution Rule
Look at some problems
18𝑥 − 3
3𝑥2 − 𝑥 + 4 4 𝑑𝑥
𝑢 = 3𝑥2 − 𝑥 + 4
Sec. 5.5: The Substitution Rule
Look at some problems
𝑥
1 − 4𝑥2 𝑑𝑥
𝑢 = 1 − 4𝑥2
Sec. 5.5: The Substitution Rule
Look at some problems
tan 𝑥 𝑑𝑥
𝑢 = cos 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
𝑠𝑖𝑛2𝑥 cos 𝑥 𝑑𝑥
𝑢 = sin 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
sin ( 𝑥)
𝑥 𝑑𝑥
𝑢 = 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
sin 𝑥 sin cos 𝑥 𝑑𝑥
𝑢 = cos 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
5𝑡 sin 5𝑡 𝑑𝑡
𝑢 = 5𝑡
Sec. 5.5: The Substitution Rule
Look at some problems
𝑠𝑒𝑐2𝑥
𝑡𝑎𝑛2𝑥𝑑𝑥
𝑢 = tan 𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
sin 2𝑥
1 + 𝑐𝑜𝑠2𝑥𝑑𝑥
𝑢 = 1 + 𝑐𝑜𝑠2𝑥
Sec. 5.5: The Substitution Rule
Look at some problems
𝑒1/𝑥
𝑥2 𝑑𝑥
𝑢 =1
𝑥
Sec. 5.5: The Substitution Rule
Ex 1: Find 2𝑥 𝑠𝑖𝑛 𝑥2 𝑑𝑥
Sec. 5.5: The Substitution Rule
Ex 2: Find −3𝑥2 2𝑥3 + 7 𝑑𝑥1
−1
Sec. 5.5: The Substitution Rule
Ex 3: Find 18𝑥 − 3
3𝑥2 − 𝑥 + 4 4 𝑑𝑥
Sec. 5.5: The Substitution Rule
Ex 4: Find tan 𝑥 𝑑𝑥
𝜋4
−𝜋3
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
• Sometimes you make u a part of the thing you are
integrating whose derivative is just a number
• The integral needs to end up simpler than the
original integral
• But!!! Don’t ever make u = x because this does
nothing!!!
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
3𝑥 − 7 𝑑𝑥
𝑢 = 3𝑥 − 7
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
𝑒4−9𝑥 𝑑𝑥
𝑢 = 4 − 9𝑥
Sec. 5.5: The Substitution Rule
Ex 5: Find 3𝑥 − 7 𝑑𝑥
Sec. 5.5: The Substitution Rule
Ex 6: Find 𝑒4−9𝑥 𝑑𝑥2
−1
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
• In general, if after making a u substitution you end
up with a simpler integral that you know how to
integrate, then do it that way
• Make sure you are able to get rid of all x terms
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
𝑥
1 + 2𝑥 𝑑𝑥
𝑢 = 1 + 2𝑥
Sec. 5.5: The Substitution Rule
Other Times You Might Use u-substitution
𝑥3 𝑥2 + 1 𝑑𝑥
𝑢 = 𝑥2 + 1
Sec. 5.5: The Substitution Rule
Ex 7: Find 𝑥
1 + 2𝑥 𝑑𝑥
Sec. 5.5: The Substitution Rule
Ex 8: Find 𝑥3 𝑥2 + 1 𝑑𝑥1
0
Chapter 6:
Applications of Integration
Section 6.5
Average Value of a Function
What does average mean? Story …
Derive formula on board …
Sec. 6.5: The Average Value of a Function
If f is integrable on [a, b] , then its average value is
Sec. 6.5: The Average Value of a Function
𝑓𝑎𝑣𝑒 =1
𝑏 − 𝑎 𝑓 𝑥 𝑑𝑥𝑏
𝑎
Ex 9: Find the average value of the function
𝑔 𝑡 =𝑡
3+𝑡2 on [1, 3]
Sec. 6.5: The Average Value of a Function
Result:
Sec. 6.5: The Average Value of a Function
Ex 10:
a) Find the average value of the function 𝑓(𝑥) =1
𝑥
on [1, 4]
b) Find the value of c guaranteed by the Mean Value
Theorem for Integrals such that 𝑓𝑎𝑣𝑒 = 𝑓(𝑐)
Sec. 6.5: The Average Value of a Function
