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

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

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

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

- ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= - ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= - ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

y = 5 + 5 x^{2} - 2 x^{3}, (1, 8)

x \to \infty lim (\frac{2}{3 x})

\int 10000 (t - 6) d t

\int 5 x sin (x) d x

14 x

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