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

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

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

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

3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= 3 ln ∣ x + 2 ∣ + 2 ln ∣ x - 2 ∣ + C

\frac{d y}{d x} = x^{2} (1 + y), y (0) = 3

x \to 0 lim (csc (x) ln (x + 1))

5 + \frac{1}{2} sin (5 π t)

y^{^{''}} - 8 y^{^{'}} + 20 y = 100 x^{2} - 91 x e^{x}

\frac{d}{d x} (x^{- \frac{1}{4}})

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