How do you use partial fractions to find the integral int (x+2)/(x^2-4x)dx?

1 Answer
Dec 28, 2016

First, factor the denominator.

x^2 -4x = x(x - 4)

A/x + B/(x - 4) = (x + 2)/((x)(x - 4))

A(x -4) + B(x) = x + 2

Ax+ Bx - 4A = x + 2

(A + B)x - 4A = x + 2

We can now write a systems of equations.

{(A + B = 1),(-4A = 2):}

Solving, we get A = -1/2 and B = 3/2.

:. The partial fraction decomposition is 3/(2(x - 4)) - 1/(2x).

This can be integrated using the rule int1/udu = ln|u| + C.

=3/2ln|x - 4| - 1/2ln|x| + C

Hopefully this helps!