Notes: Antiderivatives and Indefinite Integration
An antiderivative is a solution to a differential equation. It “undoes” the derivative.
To find an antiderivative, or indefinite integral, the function must be continuous.
dy
 f ( x)  dy  f ( x)dx  y   f ( x)dx  C
dx
* The C is called the constant of integration. It is required on all indefinite integrals
because the derivative of a constant is always 0. We call the solution with C the “general
solution”.
example:
f ( x)  x2
h( x)  x2  
g ( x)  x2  2
The derivative of each of these functions is f '( x)  2 x . If we are working backwards,
there is no way to tell if the original function had a constant without knowing a point on
the original function.
function derivative
x
n
nx
n-1
function
integral
x
x n 1
c
n 1
n
sin x
cos x
cos x
sin x+C
cos x
-sin x
sin x
-cosx+C
tan x
sec2 x
sec2 x
tan x+C
csc x
-csc x cot x
csc x cot x
-csc x+C
sec x
sec x tan x
sec x tan x
sec x+C
cot x
-csc2 x
csc2 x
-cot x + C
ex
ln x
ax
e
x
1
x
a x ln a
e
x
1
x
a
x
ex + C
ln |x|+C
ax
ln a +C
Power Rule for Integrals Examples:
1.

2
2.

1
3.
1
 x3 dx 
4.
 (3x
5.

1
x n 1
 x dx  n  1  c
n
3xdx 
 3xdx 
1dx 
 1dx 
2
4
 5 x3  7 x 2 )dx 
xdx 
x 1
dx 
x
6.

7.
 cos
sin x
dx 
2
x
8. Find the particular solution of f ( x) using the given information:
f ( x)  x2 , f (0)  6 , f (0)  3
9. A ball is thrown upward from an initial height of 80 feet with initial velocity 64 ft/sec.
Acceleration due to gravity is -32 ft/sec2. Find position s as a function of t.