Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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
 

“Rational fcn” means polynomial
polynomial



: 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”)