What is the integral of (x-3)/((x+8)(x-2)) ?

The integral of (x-3)/((x+8)(x-2)) is 11/10 ln|x+8|-1/10 ln|x-2|+C

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integral of (x-3)/((x+8)(x-2))

\int \frac{x - 3}{( x + 8 ) ( x - 2 )} d x

\frac{11}{10} ln ∣ x + 8 ∣ - \frac{1}{10} ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x - 3}{( x + 8 ) ( x - 2 )} d x

Take the partial fraction of

\frac{x - 3}{( x + 8 ) ( x - 2 )} : \frac{11}{10 ( x + 8 )} - \frac{1}{10 ( x - 2 )}

= \int \frac{11}{10 ( x + 8 )} - \frac{1}{10 ( 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{11}{10 ( x + 8 )} d x - \int \frac{1}{10 ( x - 2 )} d x

\int \frac{11}{10 ( x + 8 )} d x = \frac{11}{10} ln ∣ x + 8 ∣

\int \frac{1}{10 ( x - 2 )} d x = \frac{1}{10} ln ∣ x - 2 ∣

= \frac{11}{10} ln ∣ x + 8 ∣ - \frac{1}{10} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{11}{10} ln ∣ x + 8 ∣ - \frac{1}{10} ln ∣ x - 2 ∣ + C

a^{2} - x^{2}

h (x) = \frac{2 x}{sin ( x )}

\frac{d}{d x} ((x^{2} + x) (6 - x))

\int \frac{1}{7 x + 6} d x

(2, 4), (5, 13)

What is the integral of (x-3)/((x+8)(x-2)) ?
The integral of (x-3)/((x+8)(x-2)) is 11/10 ln|x+8|-1/10 ln|x-2|+C