7.5 Strategy for Integration
Strategy for Integration
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 applying a strategy is a knowledge of the basic integration formulas.
Strategy for Integration
In the table of Integration Formulas we have collected the integrals 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 fractions, and trigonometric substitutions can be used in place of
Formula 20.
Strategy for Integration
Strategy for Integration
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:
=
∫
sin θ cos θ dθ =∫
sin 2θ dθStrategy for Integration
∫
(sin x + cos x)2 dx =∫
(sin2x + 2 sin x cos x + cos2x) dx=
∫
(1 + 2 sin x cos x) dx2. Look for an Obvious Substitution
Try to find some function u = g(x) in the integrand whose differential du = g′(x) dx also occurs, apart from a
constant factor. For instance, in the integral
we notice that if u = x2 – 1, then du = 2x dx.
Strategy for Integration
Therefore we use the substitution u = x2 – 1 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 f(x).
(a) Trigonometric functions. If f(x) is a product of powers of sin x and cos x, of tan x and sec x, or of cot x and csc x, then we use the substitutions.
(b) Rational functions. If f is a rational function, we use the procedure involving partial fractions.
Strategy for Integration
(c) Integration by parts. If f(x) is a product of a power of x (or a polynomial) and a transcendental function (such as a trigonometric, exponential, or logarithmic
function), then we try integration by parts, choosing u and dv.
(d) Radicals. Particular kinds of substitutions are recommended when certain radicals appear.
(i) If occurs, we use a trigonometric substitution.
(ii) If occurs, we use the rationalizing substitution More generally, this sometimes works for .
Strategy for Integration
4. Try Again
If the first three steps have not produced the answer, remember that there are basically only two methods of integration: substitution and parts.
(a) Try substitution. Even if no substitution is obvious (Step 2), some inspiration or ingenuity (or even
desperation) may suggest an appropriate substitution.
(b) Try parts. Although integration by parts is used most of the time on products of the form described in Step
Strategy for Integration
(c) Manipulate the integrand. Algebraic manipulations (perhaps rationalizing the denominator or using
trigonometric identities) may be useful in transforming the integral into an easier form. These manipulations may be more substantial than in Step 1 and may
involve some ingenuity. Here is an example:
Strategy for Integration
(d) Relate the problem to previous problems. When you have built up some experience in integration, you
may be able to use a method on a given integral that is similar to a method you have already used on a previous integral. Or you may even be able to
express the given integral in terms of a previous one.
Strategy for Integration
For instance,
∫
tan2x sec x dx is a challenging integral, but if we make use of the identity tan2x = sec2x – 1,we can write∫
tan2x sec x dx =∫
sec3x dx –∫
sec x dxand if
∫
sec3x dx has previously been evaluated, then that calculation can be used in the present problem.Strategy for Integration
(e) Use several methods. Sometimes two or three
methods are required to evaluate an integral. The evaluation could involve several successive
substitutions of different types, or it might combine integration by parts with one or more substitutions.
Example 1
In Step 1 we rewrite the integral:
The integral is now of the form
∫
tanmx secnx dx with m odd.Alternatively, if in Step 1 we had written
Example 1
then we could have continued as follows with the substitution u = cos x:
cont’d
Can We Integrate All Continuous
Functions?
Can We Integrate All Continuous Functions?
The functions that we have been studied here are called elementary functions.
These are the polynomials, rational functions, power functions (xn), exponential functions (bx), logarithmic
functions, trigonometric and inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, and all
functions that can be obtained from these by the five
operations of addition, subtraction, multiplication, division, and composition.
Can We Integrate All Continuous Functions?
For instance, the function
is an elementary function.
If f is an elementary function, then f′ is an elementary
function but ∫ f(x) dx need not be an elementary function.
Consider f(x) = .
Can We Integrate All Continuous Functions?
Since f is continuous, its integral exists, and if we define the function F by
then we know from Part 1 of the Fundamental Theorem of Calculus that
Thus f(x) = has an antiderivative F, but it has been proved that F is not an elementary function.
This means that no matter how hard we try, we will never succeed in evaluating ∫ dx in terms of the functions we
Can We Integrate All Continuous Functions?
The same can be said of the following integrals:
In fact, the majority of elementary functions don’t have elementary antiderivatives.
You may be assured, though, that the integrals in the following exercises are all elementary functions.
