# Techniques of Integration

**1 Integration Tools**

In many ways, the operation of differentiation can be condensed into a black

box: once all the rules are known , one can simply create an algorithm to

analyze any expression and apply those rules as necessary to obtain the

desired result. In most cases, no creativity or ingenuity is needed to perform

differentiation. However, integration is quite the opposite. Every integral is

different and requires a different combination of techniques and methods to

derive the answer. In essence, integration is an art.

In all that follows, we shall omit including the constant of integration when

giving expressions for the antiderivatives, but the reader should always be

mindful of its implied presence.

**1.1 The Basics**

For completeness, we shall include the basic integration formulas including

the inverse power rule , and some basic trigonometric integrals .

**1.2 u-Substitution**

The method of u- substitution is a change of variables in an attempt to transform

the integrand to something that is recognizably integrable. It is here

where we first see that integration requires some adroitness. The method of

u-substitution can be summarized as follows :

1. Select of function of x (normally a portion of the
integrand) to mold

into a new variable u. I.e. make a definition of a new variable u in

terms of x . This requires some foresight in that we hope to also find

u'(x) in the integrand so that we can complete u-substitution. (see step

4.)

2. Calculate

3. Replace the expression u(x) in the integrand with the variable u wherever

appropriate.

4. Identify in the integrand and replace
with du. This is often

the stumbling point in u-substitution.

Hopefully, at the end of these steps, we end up with an integral that we

know how to compute. Just as a picture is worth a thousand words, so much

more valuable an example is than the theory:

In order to compute this integral, we notice that it is
not an integral of

any basic form and so we do not have any rules that we can immediately

apply to determine the answer. However, after staring at the page for a

minute or two, we may finally realize that so
if we define u(x)

to be ln x, then u'(x) is in the integrand and we can complete the steps in

u -substitution. Putting it all down on paper we have:

**1.3 Properties and Identities of Mathematical Functions **

There are many situations in which very nasty-looking integrals actually have

relatively simple solutions because the integrand can be greatly simplified by

using identities of functions. For example,

may look like a relatively complicated integral at first
glance. However,

by using some trigonometric identities, we can see that in fact the integrand

reduces to 1, and so this integral is trivially easy to compute. While the

above example may be an extreme case of simplification, many seemingly

untractable problems become quite manageable after some applications of

such identities. Below are a few examples of identities and properties to keep

in mind while trying to integrate expressions.

**1.4 Integration by Parts**

There are some integrals that cannot be dealt with by any of the means

dictated so far. Actually, there are many such integrals. A very powerful

tool is the integration by parts formula. The one dimensional case can be

derived from the product rule :

Rearranging terms, we obtain

The power of this method can be seen in the following way:
suppose we

are trying to integrate a product of two functions, where one function is the

derivative of something (e.g. v'). Then using this method, we essentially

transfer the derivative onto the other function u and supplement such an action

with a correction term, uv. This operation of ”transferring derivatives”

is indeed very useful in integrating a great many things. In order to illustrate

the idea, we consider an example.

No previous techniques are applicable to this example.
However, let us

consider using integration by parts. We want to ”transfer” a derivative from

one term onto another. We notice that f(x) = x is the derivative of
and

that g(x) = sin x is the derivative of −cos x. If we transfer a derivative from

f(x) to g(x), we will end up trying to integrate something proportional to

x ^{2} cos x, which is hardly better than we started off with. However,
if instead

we do the opposite, we will only have to integrate cos x, which is quite within

our capabilities. We then proceed as follows:

**1.5 Trigonometric Substitution**

When doing u-substitution, we were looking to assign a more complicated

function of x a very compact name (namely u). In this form of substitution,

we want to expand the variable x from itself into a trigonometric function.

The motivation for this is as follows: suppose we wanted to integrate something

like . None of the tricks we have learned so
far enable us to do

this. But what if the x^{2} in the denominator were a sin^{2} u
instead? Then

we could use our trigonometric identity 1 − sin^{2} x = cos^{2}
x to get rid of the

square root sign and integrate the result. We now present the mathematics

of this substitution:

Then our integral transforms as follows

A trigonometric substitution allows us to deal with terms
of the form

a−bx^{2}, bx^{2} −a, and a+bx^{2} which are under a
square root. The idea is that

we wish to coalesce the two terms into one and then get rid of the square

root. In certain cases we are even able to use trigonometric substitution to

deal with higher powers of x that may be raised to a fractional power other

than .

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