What is the integral of (3x+2)/((x-4)(x+2)) ?

The integral of (3x+2)/((x-4)(x+2)) is 7/3 ln|x-4|+2/3 ln|x+2|+C

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integral of (3x+2)/((x-4)(x+2))

\int \frac{3 x + 2}{( x - 4 ) ( x + 2 )} d x

\frac{7}{3} ln ∣ x - 4 ∣ + \frac{2}{3} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{3 x + 2}{( x - 4 ) ( x + 2 )} d x

Take the partial fraction of

\frac{3 x + 2}{( x - 4 ) ( x + 2 )} : \frac{7}{3 ( x - 4 )} + \frac{2}{3 ( x + 2 )}

= \int \frac{7}{3 ( x - 4 )} + \frac{2}{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{7}{3 ( x - 4 )} d x + \int \frac{2}{3 ( x + 2 )} d x

\int \frac{7}{3 ( x - 4 )} d x = \frac{7}{3} ln ∣ x - 4 ∣

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

= \frac{7}{3} ln ∣ x - 4 ∣ + \frac{2}{3} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{7}{3} ln ∣ x - 4 ∣ + \frac{2}{3} ln ∣ x + 2 ∣ + C

e^{t}

\int (5 x + 1) (2 x - 2) d x

\int \frac{2 x ^{2} - 5 x - 7}{x - 3} d x

\int \frac{6 x}{e ^{3 x}} d x

\int 3 e^{x} + 2 d x

What is the integral of (3x+2)/((x-4)(x+2)) ?
The integral of (3x+2)/((x-4)(x+2)) is 7/3 ln|x-4|+2/3 ln|x+2|+C