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

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

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

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

2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

\int \frac{3}{x - 2} d x = 3 ln ∣ x - 2 ∣

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 2 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 2 ∣ + C

x^{9}, at x = 1

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

f (x) = x^{2} x^{3} + 1, at x = 5

\frac{4 s}{( s ^{2} + 16 )}

x \to 0 lim (18 - x^{2})

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