unit #11 : integration by parts; using tables; average of...

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.


Use integration by parts to evaluate

∫xe4x dx.

6

Verify that your anti-derivative is correct.


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.


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

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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


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.


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


∫1

x2 + 4x + 3dx

Use the table entry you selected to evaluate the integral I.

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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


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.


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.


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].

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Example: Find the exact average of f(x) = x2 on the interval x ∈ [−2, 2]using this formula.


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.

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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?


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.

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Example: Without computing any integrals, explain why the average valueof cos(x) on x ∈ [0, π/2] must be greater than 0.5.


Can you make a general statement about the average value of decreasing func-tions which are concave down?

unit #11 : integration by parts; using tables; average of...

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