Methods of integration: Simple or partial fractions

This method aims to solve an integral in which the integrand is a rational function to decompose the fraction into simple or partial fractions, after we can decompose the radicand into simpler rational fractions is that we will be able to apply the formulas of the basic rules of integration.

Solve an integral by the method of simple fractions when there are distinct linear factors.

For this purpose I will propose to solve the following rational integral which has two different linear factors:

So far we cannot see the distinct linear factors, that is because we have to factor the polynomial of the denominator of the fraction, when we factor the polynomial of second degree we are left with:

So the integrand once having factored the polynomial of second degree that is in the denominator of the fraction would look like this:

We find the values of A and B:

To find A, we make x = 1 since B cancels out:

To find B, we make x = -2 since A cancels out:

Already having the values of A and B, A = 1 ; B = -1, we solve the integral:

As the result is the result of solving these two integrals, then we solve it by the method of substitution or change of variable:

Therefore and considering the properties of natural logarithm, in which, the logarithm of the subtraction is the logarithm of the division, the final result is:

Bibliographic Reference

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006