Strategy for Integration
As we have seen, integration is more challenging than differentiation. In finding the deriv
ative of a function it is obvious which differentiation formula we should apply. But it may
not be obvious which technique we should use to integrate a given function.
Until now individual techniques have been applied in each section. For instance, we
usually used substitution in Exercises 5.5, integration by parts in Exercises 5.6, and partial
fractions in Exercises 5.7 and Appendix G. But in this section we present a collection of
miscellaneous integrals in random order and the main challenge is to recognize which
technique or formula to use. No hard and fast rules can be given as to which method
applies in a given situation, but we give some advice on strategy that you may find useful.
A prerequisite for strategy selection is a knowledge of the basic integration formulas.
In the following table we have collected the integrals from our previous list together with
several additional formulas that we have learned in this chapter. Most of them should be
memorized. It is useful to know them all, but the ones marked with an asterisk need not be
memorized since they are easily derived. Formula 19 can be avoided by using partial frac
tions, and trigonometric substitutions can be used in place of Formula 20.
Table of Integration Formulas
Constants of integration have been omitted.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
1
7.
18.
*19.
*20.
y
dx
s
x
2
a
2
ln
x
s
x
2
a
2
y
dx
x
2
a
2
1
2
a
ln
x
a
x
a
y
dx
s
a
2
x
2
sin
1
x
a
y
dx
x
2
a
2
1
a
tan
1
x
a
y
cosh
x
dx
sinh
x
y
sinh
x
dx
cosh
x
y
cot
x
dx
ln sin
x
y
tan
x
dx
ln sec
x
y
csc
x
dx
ln csc
x
cot
x
y
sec
x
dx
ln sec
x
tan
x
y
csc
x
cot
x
dx
csc
x
y
sec
x
tan
x
dx
sec
x
y
csc
2
x
dx
cot
x
y
sec
2
x
dx
tan
x
y
cos
x
dx
sin
x
y
sin
x
dx
cos
x
y
a
x
dx
a
x
ln
a
y
e
x
dx
e
x
y
1
x
dx
ln
x
n
1
y
x
n
dx
x
n
1
n
1
1
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Once you are armed with these basic integration formulas, if you don’t immediately see
how to attack a given integral, you might try the following fourstep strategy.
1. Simplify the Integrand if Possible
Sometimes the use of algebraic manipulation or
trigonometric identities will simplify the integrand and make the method of integration
obvious. Here are some examples:
2.
Look for an Obvious Substitution
Try to find some function
in the inte
grand whose differential
also occurs, apart from a constant factor.
For instance, in the integral
we notice that if
, then
. Therefore, we use the substitu
tion
instead of the method of partial fractions.
3.
Classify the Integrand According to Its Form
If Steps 1 and 2 have not led to the
solution, then we take a look at the form of the integrand
.
(a)
Trigonometric functions.
If
is a product of powers of
and
,
of
and
, or of
and
, then we use the substitutions recom
mended in Section 5.7 and
Additional Topics: Trigonometric Integrals
.
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