Understanding Engineering Mathematics

First, note that not all products can be integrated by parts, and that integration by parts
can be useful for things that are not obvious products. We may also need to use it a number
of times, or employ it in more cunning ways (see review solutions).
The first problem we face with integration by parts is which factor to integrate first.
There are few hard and fast rules, and experience counts for a lot here. Still, we can get
a lot from our knowledge of the elementary functions:

Polynomials

Rational functions

Irrational functions

Cosine and sine

Exponentials

and their inverses. Rational and irrational (containing roots of algebraic functions) functions
are difficult to deal with when combined with other functions, so we will focus on products
of polynomials, cosine and sine and exponentials. By linearity, if we can handle a power
functionxnthen we can handle a polynomial. Also, cosine and sine are virtually equivalent
so far as calculus is concerned, so we can concentrate on sine. Finally,eαx,sin(αx)are
little different toex,sinxso we can look at just products of the three functionsxn,ex,
sinx. This gives us integrals such as:

I 1 =

∫

xnexdx

I 2 =

∫

xnsinxdx

I 3 =

∫

exsinxdx

InI 1 ,I 2 integrating or differentiatingexor sinxis neither here nor there because essen-
tially the same things results. However, integratingxnwill make things worse, while
differentiating it will eventually remove it. So forI 1 ,I 2 we start off by integratingex
or sinxso that thexnwill be differentiated. InI 3 it actually doesn’t matter which we
integrate first, we will always come back to where we started – and indeed this is the
secret to this sort of integral (see review question).
lnxand inverse functions such as sin−^1 xcan be integrated by parts by treating them
as a product with 1 and integrating the 1. For example,
∫
lnxdx=

∫

1 ×lnxdx

=xlnx−

∫

x×

1

x

dx=xlnx−x+C

Solution to review question 9.1.10

(i) We will use the integration by parts formula

∫

u

dv

dx

dx=uv−

∫

v

du

dx

dx