What is the integral of 2/(x^2-x-30) ?

The integral of 2/(x^2-x-30) is 2(-1/11 ln|x+5|+1/11 ln|x-6|)+C

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integral of 2/(x^2-x-30)

\int \frac{2}{x ^{2} - x - 30} d x

2 (- \frac{1}{11} ln ∣ x + 5 ∣ + \frac{1}{11} ln ∣ x - 6 ∣) + C

Solution steps

\int \frac{2}{x ^{2} - x - 30} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{x ^{2} - x - 30} d x

Take the partial fraction of

\frac{1}{x ^{2} - x - 30} : - \frac{1}{11 ( x + 5 )} + \frac{1}{11 ( x - 6 )}

= 2 \cdot \int - \frac{1}{11 ( x + 5 )} + \frac{1}{11 ( 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

= 2 (- \int \frac{1}{11 ( x + 5 )} d x + \int \frac{1}{11 ( x - 6 )} d x)

\int \frac{1}{11 ( x + 5 )} d x = \frac{1}{11} ln ∣ x + 5 ∣

\int \frac{1}{11 ( x - 6 )} d x = \frac{1}{11} ln ∣ x - 6 ∣

= 2 (- \frac{1}{11} ln ∣ x + 5 ∣ + \frac{1}{11} ln ∣ x - 6 ∣)

Add a constant to the solution

= 2 (- \frac{1}{11} ln ∣ x + 5 ∣ + \frac{1}{11} ln ∣ x - 6 ∣) + C

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What is the integral of 2/(x^2-x-30) ?
The integral of 2/(x^2-x-30) is 2(-1/11 ln|x+5|+1/11 ln|x-6|)+C