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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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