What is the integral of 1/(x^2+5x-14) ?

The integral of 1/(x^2+5x-14) is 1/9 ln|x-2|-1/9 ln|x+7|+C

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integral of 1/(x^2+5x-14)

\int \frac{1}{x ^{2} + 5 x - 14} d x

\frac{1}{9} ln ∣ x - 2 ∣ - \frac{1}{9} ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{1}{x ^{2} + 5 x - 14} d x

Take the partial fraction of

\frac{1}{x ^{2} + 5 x - 14} : \frac{1}{9 ( x - 2 )} - \frac{1}{9 ( x + 7 )}

= \int \frac{1}{9 ( x - 2 )} - \frac{1}{9 ( x + 7 )} 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}{9 ( x - 2 )} d x - \int \frac{1}{9 ( x + 7 )} d x

\int \frac{1}{9 ( x - 2 )} d x = \frac{1}{9} ln ∣ x - 2 ∣

\int \frac{1}{9 ( x + 7 )} d x = \frac{1}{9} ln ∣ x + 7 ∣

= \frac{1}{9} ln ∣ x - 2 ∣ - \frac{1}{9} ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{1}{9} ln ∣ x - 2 ∣ - \frac{1}{9} ln ∣ x + 7 ∣ + C

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What is the integral of 1/(x^2+5x-14) ?
The integral of 1/(x^2+5x-14) is 1/9 ln|x-2|-1/9 ln|x+7|+C