What is the integral of (x-16)/(x^2+2x-8) ?

The integral of (x-16)/(x^2+2x-8) is

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integral of (x-16)/(x^2+2x-8)

\int \frac{x - 16}{x ^{2} + 2 x - 8} d x

\frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + \frac{8}{3} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + C

Solution steps

\int \frac{x - 16}{x ^{2} + 2 x - 8} d x

Expand

\frac{x - 16}{x ^{2} + 2 x - 8} : \frac{x}{x ^{2} + 2 x - 8} - \frac{16}{x ^{2} + 2 x - 8}

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{x}{x ^{2} + 2 x - 8} d x - \int \frac{16}{x ^{2} + 2 x - 8} d x

\int \frac{x}{x ^{2} + 2 x - 8} d x = \frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1)

\int \frac{16}{x ^{2} + 2 x - 8} d x = - \frac{8}{3} (ln \frac{x + 1}{3} + 1 - ln \frac{x + 1}{3} - 1)

= \frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) - (- \frac{8}{3} (ln \frac{x + 1}{3} + 1 - ln \frac{x + 1}{3} - 1))

Simplify

\frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) - (- \frac{8}{3} (ln \frac{x + 1}{3} + 1 - ln \frac{x + 1}{3} - 1)) : \frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + \frac{8}{3} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1)

= \frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + \frac{8}{3} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1)

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 2 x - 8 + \frac{1}{6} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + \frac{8}{3} (ln \frac{1}{3} (x + 1) + 1 - ln \frac{1}{3} (x + 1) - 1) + C

x \to 6 - lim (\frac{4}{x - 6})

\frac{d}{d x} (e^{x^{2}} + 2)

\int_{- 9}^{- 8} \frac{1}{x} d x

\int 9 sin (4 x) cos (2 x) d x

x^{^{'''}} + x = 0

What is the integral of (x-16)/(x^2+2x-8) ?
The integral of (x-16)/(x^2+2x-8) is