Question

\[ \Large \int \dfrac{5x^2 - 6}{x^2 - 2x - 8}\]

Answer 1

We can do polynomial division followed by partial fraction decomposition.
$$\frac{5x^2-6}{x^2-2x-8}=5+\frac{10x+34}{x^2-2x-8}=5+2\times\frac{5x+17}{(x-4)(x+2)}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =5+2\times\left[\frac{A}{x-4}+\frac{B}{x+2}\right]$$We need to determine \(A\),\(B\) such that:$$A(x+2)+B(x-4)=5x+17\\\text{for }x=4\text{ we have }A=\frac{37}6\\\text{for }x=-2\text{ we have }B=-\frac76$$So our integral can actually be written as:$$\int\frac{5x^2-6}{x^2-2x-8}=\int\left[5+2\left(\frac{37}6\times\frac1{x-4}-\frac76\times\frac1{x+2}\right)\right]dx\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =5\int dx+\frac{37}3\int\left(\frac1{x-4}\right)dx-\frac73\int\left(\frac1{x+2}\right)dx\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =5x+\frac{37}3\ln(x-4)-\frac73\ln(x+2)+C$$

Answer 2

Nice. Thanks.

Answer 3

Note that the first step was polynomial division: