Summary Antiderivatives & Integration Review

Antiderivatives/Integration




Antiderivatives- > "undo" derivatives >
- the opposite operation of
derivatives
antiderivative
-




f(x) = 3x2 derivative
<
f(x) = x3


An antiderivative of a function f(x) is another function F(x) whose derivative

is f(x)
x(F(x)) =
f(x)
The notation for the antiderivative of f(x) is often written the
using
indefinite integral :
(f(x) dx >
- all possible functions F(x) that satisfy
F'(x) = f(x)
The Constant of Integration
Whenever you take the derivative of a function f(x) , adding a constant
(does not change the derivative :
* (f(x) + () (f(x))
=
+ Ex() = f(x) + 0 =
f(x)
The general antiderivative formula is :


=
(f(x)dX
F(x) + C
the constant of integration-
WAYSbewrittenPart,
a
must
-




The Rules
Power Rule : Px" dx =+ C
,
where ne-

1 Identify it the integrand
.
2 Increase the exponent by . . 1




. Divide by the new exponent
3 4 Add constant of integration .




Constant Rule Padx .
=
ax + C where
,
a is a constant
2
1 Recognize
. that the integrand is a constant .




Multiply the constant by x

3
. Add the constant of integration (
Sum Rule : ([f(x) + g(x)]dx f f(x)dx (g(x)dx
=
+




1
. Separate the integral of a sum into the individual integrals
2 .



Integrate each function individually into rules
you
know
.
3 Add the constants of integration which can be combined into one constant
,

2