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

The integral of (((x+1)))/((x^2-x-6)) is 1/5 ln|x+2|+4/5 ln|x-3|+C

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

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

\frac{1}{5} ln ∣ x + 2 ∣ + \frac{4}{5} ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x + 1}{x ^{2} - x - 6} : \frac{1}{5 ( x + 2 )} + \frac{4}{5 ( x - 3 )}

= \int \frac{1}{5 ( x + 2 )} + \frac{4}{5 ( x - 3 )} 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}{5 ( x + 2 )} d x + \int \frac{4}{5 ( x - 3 )} d x

\int \frac{1}{5 ( x + 2 )} d x = \frac{1}{5} ln ∣ x + 2 ∣

\int \frac{4}{5 ( x - 3 )} d x = \frac{4}{5} ln ∣ x - 3 ∣

= \frac{1}{5} ln ∣ x + 2 ∣ + \frac{4}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ x + 2 ∣ + \frac{4}{5} ln ∣ x - 3 ∣ + C

\frac{\partial}{\partial x} (5^{x^{2}})

\int \frac{4}{e ^{x}} d x

\int e^{- 11 x} d x

\frac{s + 1}{s ^{2} + s + 10}

\frac{sin ( x )}{2 x}

What is the integral of (((x+1)))/((x^2-x-6)) ?
The integral of (((x+1)))/((x^2-x-6)) is 1/5 ln|x+2|+4/5 ln|x-3|+C