by the
chain rule, Calculus
Lesson
24 Integration by Substitution
Back to Dr. Nandor's Calculus Notes Page
Back to Dr. Nandor's Calculus Page
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!)