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Intergration Essay

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Below is an essay on "Intergration" from Anti Essays, your source for research papers, essays, and term paper examples.

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

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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...

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