# Definite Integration of Rational Functions

How to solve the definite integration of rational functions problems: examples and their solutions.

## Example 1

(given) = ∫_{0}^{1} 1/(3*x* + 1) *dx*

So write the antiderivative of 1/(3*x* + 1):

(1/3) ln |3*x* + 1|.

Linear change of variable rule

And put 1 and 0 into the antiderivative.

## Example 2

Just solve the definite integration

like the indefinite integration of rational functions.

The denominator is not in linear form.

So use partial fraction decomposition

to make (constant)/(linear) form.

Set 7*x*/[(2*x* + 1)(*x* - 3)] = *A*/(2*x* + 1) + *B*/(*x* - 3).

Then compare the numerators on both sides

to find *A* and *B*.

Then *A* = 1, *B* = 3.

So 7*x*/[(2*x* + 1)(*x* - 3)] = 1/(2*x* + 1) + 3/(*x* - 3).

So (given) = ∫_{0}^{1} [1/(2*x* + 1) + 3/(*x* - 3)] *dx*.

Write the antiderivative of [1/(2*x* + 1) + 3/(*x* - 3)]:

(1/2) ln |2*x* + 1| + 3 ln |*x* - 3|.

Linear change of variable rule

And put 12 and 4 into the antiderivative.

Take the exponents out from each ln.

Logarithms of powers