What is the integral of (2x+5)/(x^2+5x+2) ?

The integral of (2x+5)/(x^2+5x+2) is

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integral of (2x+5)/(x^2+5x+2)

\int \frac{2 x + 5}{x ^{2} + 5 x + 2} d x

- \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 - \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 + ln (2 x + 5 + 17 2 x + 5 - 17) + C

Solution steps

\int \frac{2 x + 5}{x ^{2} + 5 x + 2} d x

Apply Integration By Parts

: - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) - \int - \frac{2}{17} (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) d x

= - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) - \int - \frac{2}{17} (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) d x

\int - \frac{2}{17} (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) d x = - \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 - ln (2 x + 5 + 17 2 x + 5 - 17)

= - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) - (- \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 - ln (2 x + 5 + 17 2 x + 5 - 17))

Simplify

- \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) - (- \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 - ln (2 x + 5 + 17 2 x + 5 - 17)) : - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 - \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 + ln (2 x + 5 + 17 2 x + 5 - 17)

= - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 - \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 + ln (2 x + 5 + 17 2 x + 5 - 17)

Add a constant to the solution

= - \frac{1}{17} (2 x + 5) (ln \frac{1}{17} (2 x + 5) + 1 - ln \frac{1}{17} (2 x + 5) - 1) + \frac{2}{17} x ln \frac{1}{17} (2 x + 5) + 1 - \frac{2}{17} x ln \frac{1}{17} (2 x + 5) - 1 + \frac{5}{17} ln \frac{1}{17} (2 x + 5) + 1 - \frac{5}{17} ln \frac{1}{17} (2 x + 5) - 1 + ln (2 x + 5 + 17 2 x + 5 - 17) + C

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What is the integral of (2x+5)/(x^2+5x+2) ?
The integral of (2x+5)/(x^2+5x+2) is