What is the integral of (x+22)/((x+7)(x-8)) ?

The integral of (x+22)/((x+7)(x-8)) is -ln|x+7|+2ln|x-8|+C

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integral of (x+22)/((x+7)(x-8))

\int \frac{x + 22}{( x + 7 ) ( x - 8 )} d x

- ln ∣ x + 7 ∣ + 2 ln ∣ x - 8 ∣ + C

Solution steps

\int \frac{x + 22}{( x + 7 ) ( x - 8 )} d x

Take the partial fraction of

\frac{x + 22}{( x + 7 ) ( x - 8 )} : - \frac{1}{x + 7} + \frac{2}{x - 8}

= \int - \frac{1}{x + 7} + \frac{2}{x - 8} 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{1}{x + 7} d x + \int \frac{2}{x - 8} d x

\int \frac{1}{x + 7} d x = ln ∣ x + 7 ∣

\int \frac{2}{x - 8} d x = 2 ln ∣ x - 8 ∣

= - ln ∣ x + 7 ∣ + 2 ln ∣ x - 8 ∣

Add a constant to the solution

= - ln ∣ x + 7 ∣ + 2 ln ∣ x - 8 ∣ + C

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What is the integral of (x+22)/((x+7)(x-8)) ?
The integral of (x+22)/((x+7)(x-8)) is -ln|x+7|+2ln|x-8|+C