What is the integral of (x+11)/((x-4)(x+1)) ?

The integral of (x+11)/((x-4)(x+1)) is 3ln|x-4|-2ln|x+1|+C

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integral of (x+11)/((x-4)(x+1))

\int \frac{x + 11}{( x - 4 ) ( x + 1 )} d x

3 ln ∣ x - 4 ∣ - 2 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{x + 11}{( x - 4 ) ( x + 1 )} d x

Take the partial fraction of

\frac{x + 11}{( x - 4 ) ( x + 1 )} : \frac{3}{x - 4} - \frac{2}{x + 1}

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

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

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

= 3 ln ∣ x - 4 ∣ - 2 ln ∣ x + 1 ∣

Add a constant to the solution

= 3 ln ∣ x - 4 ∣ - 2 ln ∣ x + 1 ∣ + C

y = a x + b

x^{2} = y, x^{2} = - (y - 2)

\frac{\partial}{\partial x} (x + 3 y^{\frac{1}{3}})

- y^{^{'}} + \frac{3 y}{x} = \frac{y ^{2}}{x}

\frac{d}{d x} (\frac{7 x ^{2} + 3 x + 7}{x})

What is the integral of (x+11)/((x-4)(x+1)) ?
The integral of (x+11)/((x-4)(x+1)) is 3ln|x-4|-2ln|x+1|+C