176 5. METHODS OF INTEGRATION II
Problem 5.36 Compute
Z
x dx
.x a/.x b/.x c/
dx
Problem 5.37 Compute
Z
x
2
dx
.x a/.x b/.x c/
dx
5.3 PRACTICING INTEGRATION
One of the important skills for performing integrals is figuring out which method of integration
to use. Knowledge Box 5.5 may be some help (it continues on the next page). is section is no
more than a collection of problems – but, unlike all the other sections, you cannot guess which
method is useful by looking at the section the problem appears in. Practice! Practice! Practice!
Knowledge Box 5.5
Which method of integration do I use?
1. Polynomials are integrated with the power rule, one term at a time.
2. If the thing you are integrating is the derivative of something you
recognize, use the fundamental theorem – the integral is the thing
you recognize, plus “C”.
3. Look at the known derivatives, e.g., Dx arctan.x/ D
dx
x
2
C1
. If
you find one, apply rule “b”.
4. Will an algebraic re-arrangement of the terms of the integral set up
an integral you can do?
5. Look through the examples in this chapter and in Chapter 1. Are
any of them similar enough to your problem to help?
(continued)
5.3. PRACTICING INTEGRATION 177
6. Check if there is a u-substitution that makes a simpler integral.
Remember that du needs to be in there in an appropriate form.
7. Are there natural parts for integration by parts? Remember you
may need to integrate by parts several times.
8. Is your integral packed with trig functions? Flip through Section
4.3 which has special methods for many different combinations of
trig functions. Also remember that rule “c” includes applying trig
identities to set up one of these special trig methods.
9. Does the integral contain
p
x
2
˙ a
2
or
p
a
2
˙ x
2
? Consider
trigonometric substitution. is may yield integrals covered by any
of rules “b”, “c”, or “d”.
10. Is the integral a ratio of polynomials? If the numerator is not lower
degree, divide to get a remainder that does have a lower degree top.
Integrate the resulting ratio of polynomials with partial fractions.
11. If one of these rules seems to make some progress, keep going, you
may need many steps to finish an integral.
12. Try rule “d” again. Really.
13. ere are a lot of integrals that you can’t do, there are many that
no one can do. If you are really stuck, check with someone who has
more experience than you. ey may be able to recognize impossible
integrals or integrals well above your level.
Remember that a problem may use several methods of integration to complete one intergral: a
u-substitution sets up a partial fractions decomposition that leads to a trig-substitution integral.
PROBLEMS
Problem 5.38 Pick and state a method of integration and then perform the integration for
each of the following problems.
178 5. METHODS OF INTEGRATION II
1.
Z
x
3
e
x
dx
2.
Z
x
2
.x
3
C 8/ dx
3.
Z
x
x
2
C 1
dx
4.
Z
x
x
3
1
dx
5.
Z
sin.x/ cos.x/ dx
6.
Z
sin
4
.3x/ dx
Problem 5.39 Pick and state a method of integration and then perform the integration for
each of the following problems.
1.
Z
sin.2x/e
3x
dx
2.
Z
p
25x
2
C 1 dx
3.
Z
cos.x/ sin.2x/ dx
4.
Z
ln
3
.x/ C 1
x
dx
5.
Z
csc
4
.x/ dx
6.
Z
x
2
1
x
2
C 1
dx
Problem 5.40 Pick and state a method of integration and then perform the integration for
each of the following problems.
1.
Z
dx
p
1 4x
2
2.
Z
x
p
1 9x
2
dx
3.
Z
x e
p
x
dx
4.
Z
e
x
1 Ce
2x
dx
5.
Z
e
x
e
2x
3e
x
C 2
dx
6.
Z
x ln.x/ dx
5.3. PRACTICING INTEGRATION 179
Problem 5.41 Pick and state a method of integration and then perform the integration for
each of the following problems.
1.
Z
x
2
x
4
1
dx
2.
Z
x e
3
p
x
dx
3.
Z
cos.x/
sin
2
.x/ 1
dx
4.
Z
sin.2x/
cos
2
.x/ 9
dx
5.
Z
x
5
.x
2
C 2/
4
dx
6.
Z
cos
2
.x/ sin
2
.x/
e
sin.2x/
dx
Problem 5.42 Compute
Z
.
sin.x/ C cos.x/
/
2
dx
Problem 5.43 Compute
Z
x
n
ln.x/ dx
Problem 5.44 Compute
Z
xe
n
p
x
dx
Problem 5.45 Compute
Z
.
sin.ax/ Ccos.bx/
/
e
x
dx
Problem 5.46 Compute
Z
.1 Cln.x//x
x
dx
Problem 5.47 Find a problem in this section that can be done in two ways and demonstrate
them.
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