chapter 5: integrals - greg's rio hondo math page 1 lecture notes... · u-substitution: •...
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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
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
• 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
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