taylor blog # 15

This week we were only able to tackle on major concept broken into two concepts.
These two concepts were Antiderivatives and integrals

ANTIDERIVATIVES

antiderivatives are different from an integral only because antiderivatives are classified as a general solution whereas integrals are classified as particular solutions.

Basically to find antiderivatives you apply the steps of taking a derivative backwards
or
you can memorize the formula for antiderivatives

THE FORMULA FOR FINDING AN ANTIDERIVATIVE:

Given Problem: a X ^#

A/#+1 X ^#+1
and because this is a general solution we have to account for the fact that there may or may not be a constant to account for
to do this we always add "+ c" to the end of the solved antiderivative.


EXAMPLE:

Given: 3x^2

3/2+1 X^2+1
3/3X^3
X^3

Therefore the antiderivative would be
X^3+C

For Problems with polynomials you solve piece by piece just like you would do to take the derivative of a polynomial

Example:
Given: (x+2)

1/1+1 X ^ 1+1 AND 2x
Therefore the antiderivative would be
1/2 X^2 + 2x + c



INTEGRALS


As stated earlier
integrals are different from an antiderivatives only because antiderivatives are classified as a general solution whereas integrals are classified as particular solutions.

the only component of an Integral which makes it a Particular solution is the fact that you solve for c

You still use the formula but once you solve for the antiderivative you plug in the given f(#)=0 in for X and set = 0 and solve for C and complete the solution by rewriting the antiderivative with c= # plugged in for c

Example: 1/x^2 and F'(1)=0

1/-2+1 X ^ -2+1
1/-1 X ^-1
-X^-1 +c
-1/X^1+c

plug in

-1/1 + c =0
-1 + c=0
Therefore c=1

Complete the solution

-1/x+1