Introduction to Antiderivatives
Find the derivative of each function:
3
2
4
-1
-3
1. y = 5x – 2x + 6
3. y = -3x + 5x – 2x
____________________
_________________________________
3
2. y = 7x
½
¾
¼
4. y = 12x + 8x - x
+ 6x7 – 4x2 – 5
___________________________
Examine the relationship between each problem and its answer. Could you
work backward to find the “antiderivative”?
Find the antiderivative of 8x2 + 4x ________________________________
Check by finding the derivative of your answer.
Steps for finding (using the power rule):
Derivative
1. Multiply by the exponent
2. Subtract one from the exponent
Antiderivative
1. Add one to the exponent
2. Divide by the new exponent
So, to find an antiderivative for a function f , we can often reverse the process
of differentiation. (The derivative and antiderivative are inverses)
For example, if f = x 4 , then an antiderivative of f is F =
found by reversing the power rule.
x 5 , which can be
(Notice that not only is x 5 an antiderivative of f , but so are x 5 + 4 , x 5 + 6 ,
etc. In fact, adding or subtracting any constant would be acceptable.
Therefore, we will say that C represents any constant.)
We can write the reversal of the power rule in the following way:
When n is NOT -1
**When n = -1, we have to use a natural log. We’ll learn that later!
Definition: A function F is an antiderivative or an indefinite integral of
the function f if the derivative F ' = f.
We use the notation
to indicate that F is an indefinite integral of f. Using this notation, we
have
if and only if
Evaluate the antiderivative (integral) of each.
1)
∫x8 dx
2)
∫10x8 dx
3)
∫5x4 – 3x dx
4)
∫4x-2 + 3x(1/2) dx
5)
∫x-4/4 dx
6)
∫1/4x4 dx
7)
∫3x2 – x-3 + 4x – 5 dx
8)
∫2cos x dx