What is the integral of (x-56)/(x^2+x-56) ?

The integral of (x-56)/(x^2+x-56) is

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integral of (x-56)/(x^2+x-56)

\int \frac{x - 56}{x ^{2} + x - 56} d x

2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) + \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15}) + C

Solution steps

\int \frac{x - 56}{x ^{2} + x - 56} d x

Expand

\frac{x - 56}{x ^{2} + x - 56} : \frac{x}{x ^{2} + x - 56} - \frac{56}{x ^{2} + x - 56}

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 - 56} d x - \int \frac{56}{x ^{2} + x - 56} d x

\int \frac{x}{x ^{2} + x - 56} d x = 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x))

\int \frac{56}{x ^{2} + x - 56} d x = - \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15})

= 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) - (- \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15}))

Simplify

2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) - (- \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15})) : 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) + \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15})

= 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) + \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15})

Add a constant to the solution

= 2 (\frac{1}{4} ln 4 x^{2} + 4 x - 224 + \frac{1}{60} (ln \frac{16}{15} + \frac{2}{15} x - ln - \frac{14}{15} + \frac{2}{15} x)) + \frac{56}{15} (ln \frac{2}{15} x + \frac{16}{15} - ln \frac{2}{15} x - \frac{14}{15}) + C

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What is the integral of (x-56)/(x^2+x-56) ?
The integral of (x-56)/(x^2+x-56) is