What is the integral of 1/(x^2+5x-6) ?

The integral of 1/(x^2+5x-6) is 1/7 ln|x-1|-1/7 ln|x+6|+C

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integral of 1/(x^2+5x-6)

\int \frac{1}{x ^{2} + 5 x - 6} d x

\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣ + C

Solution steps

\int \frac{1}{x ^{2} + 5 x - 6} d x

Take the partial fraction of

\frac{1}{x ^{2} + 5 x - 6} : \frac{1}{7 ( x - 1 )} - \frac{1}{7 ( x + 6 )}

= \int \frac{1}{7 ( x - 1 )} - \frac{1}{7 ( x + 6 )} 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}{7 ( x - 1 )} d x - \int \frac{1}{7 ( x + 6 )} d x

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

\int \frac{1}{7 ( x + 6 )} d x = \frac{1}{7} ln ∣ x + 6 ∣

= \frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣

Add a constant to the solution

= \frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣ + C

\frac{d}{d x} (2 c x)

\int \frac{1}{x ( ln ( x ^{4} ) ) ^{9}} d x

e^{x} y \frac{d y}{d x} = e^{- y} + e^{- 2 x - y}

3 sin (2 t)

\int - 2 x^{- 4} d x

What is the integral of 1/(x^2+5x-6) ?
The integral of 1/(x^2+5x-6) is 1/7 ln|x-1|-1/7 ln|x+6|+C