What is the integral of (x+34)/((x-6)(x+2)) ?

The integral of (x+34)/((x-6)(x+2)) is 5ln|x-6|-4ln|x+2|+C

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integral of (x+34)/((x-6)(x+2))

\int \frac{x + 34}{( x - 6 ) ( x + 2 )} d x

5 ln ∣ x - 6 ∣ - 4 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{x + 34}{( x - 6 ) ( x + 2 )} d x

Take the partial fraction of

\frac{x + 34}{( x - 6 ) ( x + 2 )} : \frac{5}{x - 6} - \frac{4}{x + 2}

= \int \frac{5}{x - 6} - \frac{4}{x + 2} 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 - 6} d x - \int \frac{4}{x + 2} d x

\int \frac{5}{x - 6} d x = 5 ln ∣ x - 6 ∣

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

= 5 ln ∣ x - 6 ∣ - 4 ln ∣ x + 2 ∣

Add a constant to the solution

= 5 ln ∣ x - 6 ∣ - 4 ln ∣ x + 2 ∣ + C

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What is the integral of (x+34)/((x-6)(x+2)) ?
The integral of (x+34)/((x-6)(x+2)) is 5ln|x-6|-4ln|x+2|+C