詳解：分母可分解

\({x^2} + 3x - 10 = (x - 2)(x + 5)\)

\(\int {\frac{{5x + 2}}{{{x^2} + 3x - 10}}\;dx}  = \int {\frac{{5x + 2}}{{(x - 2)(x + 5)}}\;dx} \)

先強迫分解為兩個一次式

\(\int \frac{5x+2}{(x-2)(x+5)}dx = \int \frac{?}{x-2}+\frac{?}{x+5}dx\)

利用快速判斷係數法，求？中的數

\(\int \frac{?}{x-2}+\frac{?}{x+5}dx=\int \frac{\frac{12}{7}}{x-2}+\frac{\frac{-23}{-7}}{x+5}dx\)

分解成兩個簡單的一次式積分

\(\int {\frac{{{\textstyle{{12} \over 7}}}}{{x - 2}}}  + \frac{{{\textstyle{{ - 23} \over { - 7}}}}}{{x + 5}}dx = \frac{{12}}{7}\ln (x - 2) + \frac{{23}}{7}\ln (x + 5) + C\)