The Definite Integral

Remember the difference between the indefinite integral and the definite integral:
Indefinite

• will have a "+ C"
• answer will be a function
• no upper/lower bound on integral symbol

Definite

• no "+ C"
• answer will be a NUMBER
• it has upper/lower bounds on integral symbol

Steps to solving:

1. Find the integral

2. Plug in upper bound

3. Plug in lower bound

4. Subtract step #3 from step #2

Website with worked examples and good explanations!

Remember what we are REALLY doing...calculating definite integrals is the "easy" way to add up all those infinite rectangles and get the TRUE area under a curve.

Change of Variable...the Dirty Integral

As we attempt to undo chain rule-esque intergrals remember the following:

1. Pick a good "u".  It's is usually the "inside stuff"...but doesn't have to be. 
2. Find "du/dx"...take the derivative of u.
3. Your "dx" and anything else left over in the intergral should show up once you find "du/dx".  Sub that into your intergral.  You've now successfully changed your variables from "x" to "u".  Hopefully, the new intergral is simplier.  Take the integral.
4. Resub to get "x" in your answer.

Easy as pi... :)

Differential Equations

Today we discussed solving equations involving derivatives and anti-derivatives.  Most importantly...we use some given information to solve for the dreaded "arbitrary constant"...

1. You will be given a derivative or 2nd derivative (f'(x) or f''(x))...use techniques of integration to find either f(x) or f'(x)...there will be a "+C" in your answer.
2. Use the information given to you to solve for C.
3. Repeat if necessary...

Indefinite Integrals

And so we finally learn how to "undo" the derivative.  Remember, reversing the power requires two steps:

1. Raise the power by 1.
2. Divide the coefficient by the new power.

That's it.  :)
And don't forget your 6 trig integrals:

1.)

2.)

3.)

4.)

5.)

6.)