Review of Techniques of Integration

Strategy

• Identify the type of integral you have.

• Try using the simplest technique for your integral.

• If that doesn’t work, try the next-level method
(might involve more work, but is more likely to
work).

• You may have to break-up the integral into several
parts and use a different method for each
part.

u-Substitution

Goal:
Reduce the integral to one of the 10 basic
integrals in the table.

Works:
If the integrand is nice enough to have exactly
the chain rule form:


Examples:

Integration by parts

Goal:
Figure out the integral of a product of two functions
(polynomials, exponentials, logs, trig, inverse
trig, etc).

Works: For all the times when u-substitution fails

Notes:
Needed for (Use Trig-Tricks for
most other trig integrals)

Examples:

Trig Tricks

Goal: Integrals of combinations of trig functions
Examples:

The 4 steps

Check these steps (in order!) to simplify your integral
as much as needed:

1: If you have an integral of a product of
trig functions with different angles (Ex: 3x ≠
4x) you MUST use trig identities to break it
up into a sum of different terms before you can
make any progress:


2: If one is odd, u = other This gives the right
u-substitution for integrals
(with positive or zero powers). You will also
need to use cos² x + sin² x = 1.. Example:
then , so u = sin x and
use cos² x = 1 − u² to get

3:  If both powers are even,
then you need to use the half-angle formulas:



4: SET TOS Use sec² x = 1 + tan² x with
• SET:
• TOS:
Ex:

Square roots

• If you have a then let u = so
u² = ax+b and 2u du = a dx and x = (u²−b)/a
• If you have a then
• If you have a then
• If you have a then
• If you have a then First,
complete the square, then let u = x−d and
look again, example:



• These work with positive or negative powers of
the roots and (quadratic)^±k too.

Rational Functions
 

: Complete the square and use
square roots guide.


Everything else needs Partial Fractions:

1. NO Improper fractions: Convert .
You MUST do this first! (Use long division or
synthetic division)

2. Factor numerator, denominator, cancel stuff

3. Expand out as a sum of partial fractions (each
will be one of the “easy ones”)