Calculus 24

Calculus

Lesson 24 Integration by Substitution

Since by the chain rule,

we also know that by definition:

So, the key is to look for a function that looks like it the derivative of another

function multiplied by that function. Repeat: LOOK FOR BOTH A FUNCTION

AND ITS DERIVATIVE IN THE INTEGRAL!!!!

DON'T PANIC!!

Example: Note that .

So use "u substitution":

Then integrate.....

Other examples: You can check each by differentiating!!

(don't need the chain rule for this so don't use it!)

For definite integrals, you must change limits!!! (either into u or just use a to b)

Example:

Other examples: Note for the top one the bottom limit cannot=0.

If you are stuck and cannot find a function and derivative, BUT you still see

something that looks like it might be easy to integrate, try u-sub anyway!

Example:

Two more tricks:

For even functions, area on either side of zero is equal, so

draw picture

For odd functions, the areas will cancel, so

draw picture

Example:

(Check that before you make any conclusions!)

On to Lesson 25 - Numerical Integration