What is the integral of ((3x+11))/(x^2-x-6) ?

The integral of ((3x+11))/(x^2-x-6) is -ln|x+2|+4ln|x-3|+C

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integral of ((3x+11))/(x^2-x-6)

\int \frac{( 3 x + 11 )}{x ^{2} - x - 6} d x

- ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{3 x + 11}{x ^{2} - x - 6} d x

Take the partial fraction of

\frac{3 x + 11}{x ^{2} - x - 6} : - \frac{1}{x + 2} + \frac{4}{x - 3}

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

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

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

= - ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣

Add a constant to the solution

= - ln ∣ x + 2 ∣ + 4 ln ∣ x - 3 ∣ + C

x + 11

f (x) = \frac{4 x}{x + 2}, at x = 2

\frac{\partial}{\partial x} (\frac{y ^{2}}{x ^{2} + y ^{2}})

x^{3} - 11 x^{2} + 18 x, - x^{3} + 11 x^{2} - 18 x

\int \frac{- ln ( x )}{2} d x

What is the integral of ((3x+11))/(x^2-x-6) ?
The integral of ((3x+11))/(x^2-x-6) is -ln|x+2|+4ln|x-3|+C