Question

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{x-12}{x^{2}-4 x} d x$$

Short answer

The solution to the integral \(\int \frac{x-12}{x^{2}-4 x} d x\) is \(4 \ln|x| - \ln|x-4| + C\)

Step-by-step solution

Factorize the denominator

Factorize the denominator \(x^{2}-4 x\) as \(x(x-4)\)

Partial Fraction Decomposition

Rewrite the fraction \(\frac{x-12}{x^{2}-4 x}\) as \(\frac{A}{x} + \frac{B}{x-4}\) where A and B are constants to be determined. Solving for A and B gives us A = 4 and B = -1.

Integrate the decomposed fraction

Integrate \( \int \frac{4}{x} dx - \int \frac{1}{x-4} dx \) which gives us \(4 \ln|x| - \ln|x-4| + C\)