Definite Integration of Rational Functions

Definite Integration of Rational Functions

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

Example 1

Find the given integral. The integral, from 0 to 1, of [dx/(3x + 1)] dx

(given) = ∫01 1/(3x + 1) dx

So write the antiderivative of 1/(3x + 1):
(1/3) ln |3x + 1|.

Linear change of variable rule

And put 1 and 0 into the antiderivative.

Example 2

Find the given integral. The integral, from 4 to 12, of [7x/(2x + 1)(x - 3)] dx

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

Then compare the numerators on both sides
to find A and B.

Then A = 1, B = 3.

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

So (given) = ∫01 [1/(2x + 1) + 3/(x - 3)] dx.

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