The Integral: the Anti-derivativeThe Integral: the Anti-derivative

By: Ashley WangBy: Ashley Wang

Math Analysis Honors: Period 4Math Analysis Honors: Period 4

November 2007November 2007

First StepFirst Step

Simplify, if possible.Simplify, if possible. This will make things a lot easier.This will make things a lot easier.

∫ ∫ (7x+2x+3)(7x+2x+3)88

∫ ∫ (9x+3)(9x+3)88

Second StepSecond Step

Make your guess.Make your guess. Raise the exponent– of the entire problem– to Raise the exponent– of the entire problem– to

one number higher.one number higher.

G: G: (9x+3)(9x+3)99

Third StepThird Step

Check your work.Check your work. Do this by setting the derivative of your guess Do this by setting the derivative of your guess

equal to the original problem.equal to the original problem.

C: 9 C: 9 (9x+3)(9x+3)8 *8 * (9) = (9x+3) (9) = (9x+3)88

==11/8181

HINT: Remember the package rule!HINT: Remember the package rule!

Fourth StepFourth Step

Plug the Plug the into the guess to get the final answer. into the guess to get the final answer.

11//81 81 (9x+3)(9x+3)9 9 11//81 81 (9x+3)(9x+3)9 9

A: A: 11//81 81 (9x+3)(9x+3)9 9

THIS is the integral of the problem.THIS is the integral of the problem.

Fifth StepFifth Step

You could always double-check the answer, You could always double-check the answer, just to be absolutely sure.just to be absolutely sure.

It doesn’t hurt!It doesn’t hurt!

So, just take the derivative of the final answer So, just take the derivative of the final answer and see if it equals the original problem.and see if it equals the original problem.

If not, go back and check your work.If not, go back and check your work.

Fifth Step continued…Fifth Step continued…

A: A: 11//81 81 (9x+3)(9x+3)9 9

A’: 9 * A’: 9 * 11//81 81 * * (9x+3)(9x+3)88 * (9) * (9)

((11//99) * (9) * ) * (9) * (9x+3)(9x+3)88 (9x+3)(9x+3)88

Yup, the answer is perfecto!Yup, the answer is perfecto!

THE ENDTHE END

Thank you Mr. Wiencek for watching my Thank you Mr. Wiencek for watching my presentation and…presentation and…

THANK YOU in advance, for giving me an THANK YOU in advance, for giving me an ‘‘AA’ in this class for the semester!!’ in this class for the semester!!

Seriously!Seriously!