What is the integral of (7x+19)/((x-3)(x+5)) ?

The integral of (7x+19)/((x-3)(x+5)) is 5ln|x-3|+2ln|x+5|+C

integral of (7x+19)/((x-3)(x+5))

\int \frac{7 x + 19}{( x - 3 ) ( x + 5 )} d x

5 ln ∣ x - 3 ∣ + 2 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{7 x + 19}{( x - 3 ) ( x + 5 )} d x

Take the partial fraction of

\frac{7 x + 19}{( x - 3 ) ( x + 5 )} : \frac{5}{x - 3} + \frac{2}{x + 5}

= \int \frac{5}{x - 3} + \frac{2}{x + 5} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{5}{x - 3} d x + \int \frac{2}{x + 5} d x

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

\int \frac{2}{x + 5} d x = 2 ln ∣ x + 5 ∣

= 5 ln ∣ x - 3 ∣ + 2 ln ∣ x + 5 ∣

Add a constant to the solution

= 5 ln ∣ x - 3 ∣ + 2 ln ∣ x + 5 ∣ + C

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What is the integral of (7x+19)/((x-3)(x+5)) ?
The integral of (7x+19)/((x-3)(x+5)) is 5ln|x-3|+2ln|x+5|+C