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The Integral
1 Antiderivatives and the Indefinite Integral
Suppose that we are given the function and we are asked to find its antiderivative. We must find a function whose derivative is It is easy to see that one such function isNotice that antiderivatives are not unique, however, since is also an antiderivative of as is In fact, any function , where C is a real number constant, is an antiderivative of Another way of saying this is to say that any function whose derivative is given by the constant must have the form In general, any two functions with the same derivative will differ only by a constant. That is, if is an antiderivative of then is also an antiderivative for any constant
Definition: A function is called an antiderivative of if 
Notation: If then is called the general antiderivative of The process of finding antiderivatives is called integration.We write to denote the general antiderivative of a function is called the indefinite integral of the function with respect to and is called an integral sign. The function is called the integrand and is called the constant of integration. In summary,
Example 1
Find the general antiderative of
Solution: We need to find a function whose derivative is Since is the derivative of we see that an antiderivative of is All other antiderivatives of will have the form
We can check this answer by taking the derivative of
Example 2
Evaluate
Solution: We are being asked to find the indefinite integral of the function with respect to That is, we must find an antiderivative of Since
the derivative of is we see that
Again, we can check the answer by differentiating:
2 Basic Integration Formulas
Since integration or antidifferentiation is the reverse process of differentiation, we can use the derivative rules in reverse to find indefinite integrals. Not all of the differentiation formulas that we learned in Lesson 6 are easy to state in reverse. The following list gives the basic integration formulas.
1. The general antiderivative of a constant is 

2. The general antiderivative of a power function is provided 
(Add 1 to the power and divide by the new power.) 
3. An antiderivative of a sum (or difference) of terms is the sum (or difference)of the antiderivatives. 

4. An antiderivative for a constant times a function is the constant times the antiderivative of the function. 

5. The general antiderivative of is 

6. The general antiderivative for is 

7. The general antiderivative for is 

The following examples illustrate how the Integration formulas work:
Example 1 Evaluate
Solution Using formula 2 above, we get
Since integration is the reverse process of differentiation, we can check this answer by taking the derivative of our answer.
Since
is the integrand in our original problem, we know that we have found the correct answer.
Example 2 Evaluate
Solution
Again, we can check this answer by differentiation
Example 3 Evaluate
Solution
Check by differentiating:
Example 4 Evaluate
Solution check by differentiating:
Example 5 Evaluate
Solution
Check the answer by differentiating:
Example 6 Evaluate
Solution
Check by differentiating:
3 The Definite Integral
The definite integral from a to b of of a continuous function
with respect to
is
where
Notice that we use the same notation as for the indefinte integral except that we add the numbers
and
at the bottom and top of the integral sign. The numbers
and
are called the limits of integration,
is the integrand, and
is simply notation for "with respect to x". So,
is read as "the definite integral of
with respect to
Example 7
Evaluate .
Solution
First we need to find an antiderivative for The antiderivative is (we will choose to let Then
Example 8
Evaluate .
Solution
An antiderivative for is So,