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Popular >

integral of (x+8)/(x^2+x-6)

Solution

\int \frac{x + 8}{x ^{2} + x - 6} d x

Solution

2 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - \frac{8}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

Solution steps

\int \frac{x + 8}{x ^{2} + x - 6} d x

Expand

\frac{x + 8}{x ^{2} + x - 6} : \frac{x}{x ^{2} + x - 6} + \frac{8}{x ^{2} + x - 6}

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} + x - 6} d x + \int \frac{8}{x ^{2} + x - 6} d x

\int \frac{x}{x ^{2} + x - 6} d x = 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x))

\int \frac{8}{x ^{2} + x - 6} d x = - \frac{8}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

= 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - \frac{8}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5})

Add a constant to the solution

= 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 24 + \frac{1}{20} (ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x)) - \frac{8}{5} (ln \frac{2}{5} x + \frac{6}{5} - ln \frac{2}{5} x - \frac{4}{5}) + C

Popular Examples

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\int \frac{- 4 x + 8}{( x ^{2} + 1 ) ( x - 1 )} d x

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\int e^{- t} sin (- 2 t) d t

integral of sqrt(x^2-6)

\int x^{2} - 6 d x

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integral of (x^2+3x+2)/(x(x^2+1))

\int \frac{x ^{2} + 3 x + 2}{x ( x ^{2} + 1 )} d x

Frequently Asked Questions (FAQ)

What is the integral of (x+8)/(x^2+x-6) ?
The integral of (x+8)/(x^2+x-6) is