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Unit #11 : Integration by Parts; Using Tables; Average of a Func-tion
Goals:
• Learning integration by parts.
• Using tables of integrals.
• Computing the average value of a function.
Reading: Sections 7.2, 7.3, 5.3
2
Integration by Parts
Reading: Section 7.2So far in studying integrals we have used
• direct anti-differentiation, for relatively simple functions, and
• integration by substitution, for some more complex integrals.
However, there are many integrals that can’t be evaluated with these techniques.
Try to find
∫xe4x dx.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 3
This particular integral can be evaluated with a different integration technique,integration by parts. This rule is related to the product rule for derivatives.Expand
d
dx(uv) =
Integrate both sides with respect to x and simplify.
Express
∫udv
dxdx relative to the other terms.
4
Integration by Parts
For short, we can remember this formula as∫udv = uv −
∫vdu
Integration by parts:
• Choose a part of the integral to be u, and the remaining part to be dv.
• Differentiate u to get du.
• Integrate dv to get v.
• Replace
∫u dv with uv −
∫vdu.
• Hope/check that the new integral is easier to evaluate.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 5
Use integration by parts to evaluate
∫xe4x dx.
6
Verify that your anti-derivative is correct.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 7
Guidelines for selecting u and dv
• Try to select u and dv so that either
– u′ is simpler than u or
–∫dv is simpler than dv
• Ensure you can actually integrate the dv part by itself
8
Example: Find
∫x cosx dx.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 9
When using integration by parts to evaluate definite integrals, you need to applythe limits of integration to the entire anti-derivative that you find.
Example: Evaluate
∫ π
0
x sin 4x dx
10
Don’t forget that dv does not require any other factors besides dx. That can helpwhen there is only a single factor in the integrand.
Example: Find
∫2
1
ln x dx
Unit 11 – Integration by Parts; Using Tables; Average of a Function 11
There are some classical problems that can be solved by integration by parts, butnot in the direct way we have seen so far.
Example: Integrate by parts twice to find
∫cos(x)ex dx.
12
Tables of Integrals
Reading: Section 7.3
We have seen a variety of integration types by now. If you did enough integrals,using substitutions or integration by parts, you would begin to see patterns; a ta-ble of integrals formalizes these patterns, letting you evaluate integrals withoutresorting to a specific technique.The inside back cover of the textbook contains a short table of integrals. Longertables may be found in other calculus books or reference books in the library.These tables are not meant to be memorized. If an integral on a
test or an exam requires you to use a table, it will be provided.
To make the tables more generally useful, you need to be able to adapt the integralyou are given to match the form in the table.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 13
Example: Consider the integral
I =
∫x5 ln(3x)dx
In the table of integrals, the entry that most closely resembles this is∫xn lnx dx =
1
n + 1xn+1 ln x−
1
(n + 1)2xn+1 + C
Find a substitution that will change I into a form matching the table entry.Use this substitution and the table entry to evaluate the integral.
14
Question: Consider the integral I =
∫1
x2 + 4x + 3dx
Which of the integral table entries will help us evaluate this integral?
1.
∫1
x2 + a2dx 2.
∫bx + c
x2 + a2dx
3.
∫1
(x− a)(x− b)dx 4.
∫cx + d
(x− a)(x− b)dx
Unit 11 – Integration by Parts; Using Tables; Average of a Function 15
∫1
x2 + 4x + 3dx
Use the table entry you selected to evaluate the integral I.
16
Example: Now consider the very similar integral
I2 =
∫1
x2 + 4x + 13dx
Which form of the integrals below can we make our integral closest to?
1.
∫1
x2 + a2dx 2.
∫bx + c
x2 + a2dx
3.
∫1
(x− a)(x− b)dx 4.
∫cx + d
(x− a)(x− b)dx
Unit 11 – Integration by Parts; Using Tables; Average of a Function 17
I2 =
∫1
x2 + 4x + 13dx
Complete the square in the denominator (see Example 8 on page 350) to putthe integral in a form to which the table can be applied.
18
Example: Use the tables to find
∫cos5(10x)dx.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 19
Applications of Integrals - Average Value
Example: If the following graph describes the level of CO2 in the air in agreenhouse over a week, estimate the average level of CO2 over that period.
0 1 2 3 4 5 6 7
350
400
450
Time (Days)
CO
2 (p
pm)
Give the units of the average CO2 level.
20
Example: Sketch the graph of f(x) = x2 from x = −2 to 2, and estimatethe average value of f on that interval.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 21
What makes the single “average” f value different or distinct from otherpossible f values?
Use this property to find a general expression for the average value of f(x)on the interval x ∈ [a, b].
22
Example: Find the exact average of f(x) = x2 on the interval x ∈ [−2, 2]using this formula.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 23
Example: The temperature in a house is given by H(t) = 18 + 4 sin(πt/12),where t is in hours and H is degrees C. Sketch the graph of H(t) from t = 0to t = 12, then find the average temperature between t = 0 and t = 12.
24
Example: You are told that the average value of f(x) over the interval
x ∈ [0, 3] is 5. What is the value of
∫3
0
f(x) dx?
Unit 11 – Integration by Parts; Using Tables; Average of a Function 25
If f(x) is even, what is
∫3
−3
f(x) dx? Sketch such a possible function.
If f(x) is odd, what is
∫3
−3
f(x) dx? Sketch such a possible function.
26
Example: Without computing any integrals, explain why the average valueof cos(x) on x ∈ [0, π/2] must be greater than 0.5.
Unit 11 – Integration by Parts; Using Tables; Average of a Function 27
Can you make a general statement about the average value of decreasing func-tions which are concave down?