Honors Finite – Sec. 11.1- Antiderivatives and Indefinite Integrals
Of x 2  5?
What is the derivative of x 2 ?
Of x 2  8?
Notice they all have the same derivative. All of the above expressions are
antiderivatives of 2x. The most general antiderivative of 2x is x 2  C , where C is any
constant. The most general antiderivative is called the indefinite integral .
 2xdx  x
2
C
An indefinite integral can be checked by differentiation.
 f ( x)dx  g ( x)  C if and only if g '( x)  f ( x).
Power Rule for Integration
1
 x dx  n  1 x
n
n 1
 C , n  1 ,
To integrate x to a power, add 1 to the power and divide by the new power.
Example 1:
 x dx
4
Example 2:
What happens when you try this problem?
1
3
 x dx
x
1
Example 3:  1dt
dx
Other Rules:
1. The integral of a sum is the sum of the integrals.
2. The integral of a constant times a function is the constant times the
integral of the function.
3. The integral of a constant is the constant times x + C.
3
Example 4:
 (14
Example 5:
  x  2
Example 6:
4 x4  4 x2  x
dx

x
Example 7:
x2 1
 x  1 dx
4
x3 
2
4
x3
)dx
dx
4
3
1
2
Example 8: A company’s marginal cost function is MC  21x  6 x  50 , where x is the
number of units, and fixed costs are $3000. Find the cost function.