Exam (elaborations)
7.1 Integration by parts
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7.1 Integration by parts
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IntegrationINTEGRATION BY PARTS
Graham S McDonald
A self-contained Tutorial Module for learning
the technique of integration by parts
● Table of contents
● Begin Tutorial
c 2003 g.s.mcdonald@salford.ac.uk
, Table of contents
1. Theory
2. Usage
3. Exercises
4. Final solutions
5. Standard integrals
6. Tips on using solutions
7. 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
dv d du
u = (uv) − v
dx dx dx
Now, integrating both sides with respect to x results in
Z Z
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 dx ). The other factor is taken to
dv
be dx (on the right-hand-side only v appears – i.e. the other factor
integrated with respect to x).
Toc JJ II J I 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:
sin bx
xn · xn ·eax dx
R R
(i) or dx (ii)
cos bx
↑ ↑ ↑ ↑
dv dv
u dx u dx
sin bx
xr · ln (ax) dx eax ·
R R
(iii) (iv) or dx
cos bx
↑ ↑ ↑ ↑
dv dv
dx u u dx
where a, b and r are given constants and n is a positive integer.
Toc JJ II J I Back
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