# 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 cos

^{2}x + sin

^{2}x = 1.. Example:

then , so u = sin x and

use cos

^{2}x = 1 − u

^{2}to get

3: If both powers
are even,

then you need to use the half-angle formulas:

4: **SET TOS** Use sec^{2} x = 1 + tan^{2} x with

• **SET:
**•

**TOS**:

Ex:

**Square roots**

• If you have a
then let u =
so

u^{2} = ax+b and 2u du = a dx and x = (u^{2}−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:

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”)