Question

\(\int \frac{x^{2}+x-1}{x^{2}-x} d x\)

Short answer

The evaluated integral is \(x + 2x - ln|x-1|+ C\).

Step-by-step solution

Polynomial Division

First perform polynomial division to simplify the integral \(\int \frac{x^{2}+x-1}{x^{2}-x} d x\). After dividing \(x^{2}+x-1\) by \(x^{2}-x\) you get \(1+\frac{2x-1}{x^{2}-x}\), therefore the integral becomes \(\int 1+\frac{2x-1}{x^{2}-x} dx\).

Decompose Integral

The integrand function is now in a form where it can be decomposed into more manageable sub-integrals. Decomposing the integral yields \(\int 1 dx + \int \frac{2x-1}{x^{2}-x} dx\).

Simplify Integral terms

The second integral term \(\int \frac{2x-1}{x^{2}-x} dx\) can be simplified into \(\int 2 dx - \int \frac{1}{x-1} dx\). Therefore, our whole integral becomes \(\int 1 dx + \int 2 dx - \int \frac{1}{x-1} dx\).

Calculate Each Integral

Calculated each integral separately: \(\int 1 dx = x\), \(\int 2 dx = 2x\), \(\int \frac{1}{x-1} dx = ln|x-1|\).

Combine Results

Combine the results of the computed integrals \(\int 1 dx + \int 2 dx - \int \frac{1}{x-1} dx\) to get the final result \(x + 2x - ln|x-1|+ C\).