$\int \frac{x + 5}{3 x^{2} + 13 x - 10} d x$

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Solution

$I = \int \frac{x + 5}{3 x^{2} + 13 x - 10} d x = \int \frac{x + 5}{3 x^{2} + 15 x - 2 x - 10} d x = \int \frac{x + 5}{3 x (x + 5) - 2 (x + 5)} d x = \int \frac{x + 5}{(3 x - 2) (x + 5)} d x$

$= \int \frac{x + 5}{(3 x - 2) (x + 5)} d x = \int \frac{1}{3 x - 2} d x ∴ I = \frac{1}{3} \ln |3 x - 2| + c$

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