INTEGRATION
Definition : A function F(x) is the anti-derivative
of a function f(x) if
for all x in the domain of f(x).
Then the indefinite integral of f with respect
to x is written as
where c is an arbitrary constant. It is read
integral of f(x) with respect to x.
f is called the integrand and x is called the
variable of integration.
Examples:
Note that the derivative of the right side is equal
to x.
Table of Formulas:
Examples:
Check your answer by differentiating the right
side and make sure it the integrand.
Note: Why cant we find the integral of 1/x ?
Note: we learn to evaluate such integrals with
a substitution.
Now follow the above example.
Integration by Substitution:
Note: After a substitution, the integral should
look like one of the seven integrals in the above table of formulas with
x replaced by u.
Examples:
Let 2x 1 = u
Then 2dx = du
dx = ½ du
Note: Why does the substitution not work for the following integral:
Substitute 7x 6 = u
7 dx = du
dx = 1/7 du
Note: Check your answer by differentiating the right
side.
The integral takes the form:
Substitute for u to get the answer.
Question: Will the substitution work for the
following?
Evaluate the integrals:
What do you think the substitution will be?
sin3x = u or cos 3x = u.
Will any other substitution work in this case?
Now you guess the substitution.
Use 
Substitute 1+sin 3x = u
Try a substitution similar to #13.
This can be done without a substitution.
Which substitution will work in this case?
Substitute 3 2 cosx = u.
The integral becomes:
Integration Formulas for Exponential and Logarithmic Functions:
Examples:
Evaluate the following integrals:
Note: Verify your answers by checking that the derivative of the answer is the integrand.
Let 4x 3 = u
4dx = du
Check your answer by verifying that the derivative of the right side is equal to the integrand.
Let 2 5x = u
-5dx = du
The integral becomes:
Now substitute u = 2 5x.
The integral becomes:
