An Integration Strategy
for the Stout Hearted!
Do you have
an integral?
Uh…. then why
are you here?
No
Yes
Can you simplify
the integrand?
Look for an obvious
substitution.
I got nothing.
What is the form
of the integrand?
Look for a 𝑢-substitution
where part of the
integrand is 𝑑𝑢 and the
rest can be rewritten
with trig identities in
terms of 𝑢
Radicals
Integration
by parts
Rational
function
Products of trig
functions
If improper, divide.
Otherwise, try
 Partial fractions
 Add an arbitrary 0
 Complete the square

Most often used with a
power of 𝑥 and a
transcendental function.
Other cases exist as well.

±𝑥 2 ± 𝑎2 , use trig
substitution
𝑛
𝑎𝑥 + 𝑏 , try
𝑢=
𝑛
𝑎𝑥 + 𝑏
No luck!
Manipulate
the integrand
 Multiply by the
conjugate
 Rationalize
 Add an arbitrary 0
 Use a trig identity
Try parts again
Sometimes parts
works with a single
function.
Try a non-obvious
substitution
Sometimes just trying a
substitution can yield
results. Desperate times….
Still no luck!
You’re screwed.
All hail Wolfram Alpha.
This document is licensed under a Creative Commons AttributionNoncommercial-Share Alike 3.0 United States License. Dan Kernler, 2015.