Integration

INTEGRATION BY PARTS

Graham S McDonald

A self-contained Tutorial Module for learning

the technique of integration by parts

q Table of contents

q Begin Tutorial

c 2003 g.s.mcdonald@salford.ac.uk

Table of contents

1.

2.

3.

4.

5.

6.

7.

Theory

Usage

Exercises

Final solutions

Standard integrals

Tips on using solutions

Alternative notation

Full worked solutions

Section 1: Theory

3

1. Theory

To diﬀerentiate a product of two functions of x, one uses the product

rule:

d

dv

du

(uv ) = u

+

v

dx

dx dx

where u = u (x) and v = v (x) are two functions of x. A slight

rearrangement of the product rule gives

d

du

dv

=

(uv ) −

v

u

dx

dx

dx

Now, integrating both sides with respect to x results in

dv

du

u dx = uv −

v dx

dx

dx

This gives us a rule for integration, called INTEGRATION BY

PARTS, that allows us to integrate many products of functions of

x. We take one factor in this product to be u (this also appears on

the right-hand-side, along with du ). The other factor is taken to

dx

dv

be dx (on the right-hand-side only v appears – i.e. the other factor

integrated with respect to x).

Toc

Back

Section 2: Usage

4

2. Usage

We highlight here four diﬀerent types of products for which integration

by parts can be used (as well as which factor to label u and which one

dv

to label dx ). These are:

(i)

(iii)

sin bx

or

xn ·

dx (ii)

cos bx

↑

↑

dv

u

dx

xr · ln (ax) dx

↑

dv

dx

↑

u

(iv)

xn ·eax dx

↑↑

dv

u dx

sin bx

or

eax ·

dx

cos bx

↑

↑

dv

u

dx

where a, b and r are given constants and n is a positive integer.

Toc

Back

Section 3: Exercises

5

3. Exercises

Click on Exercise links for full worked solutions (there are 14 exercises in total)

Exercise 1.

x cos x dx

Exercise 2.

x2 sin x dx

Exercise 3.

xex dx

Exercise 4.

x2 e4x dx

Exercise 5.

x2 ln x dx

q Theory q Integrals...