What is the integral of (2x+9)/(x^2+9x+2) ?

The integral of (2x+9)/(x^2+9x+2) is

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integral of (2x+9)/(x^2+9x+2)

\int \frac{2 x + 9}{x ^{2} + 9 x + 2} d x

- \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 - \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 + ln (2 x + 9 + 73 2 x + 9 - 73) + C

Solution steps

\int \frac{2 x + 9}{x ^{2} + 9 x + 2} d x

Apply Integration By Parts

: - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) - \int - \frac{2}{73} (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) d x

= - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) - \int - \frac{2}{73} (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) d x

\int - \frac{2}{73} (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) d x = - \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 - ln (2 x + 9 + 73 2 x + 9 - 73)

= - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) - (- \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 - ln (2 x + 9 + 73 2 x + 9 - 73))

Simplify

- \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) - (- \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 - ln (2 x + 9 + 73 2 x + 9 - 73)) : - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 - \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 + ln (2 x + 9 + 73 2 x + 9 - 73)

= - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 - \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 + ln (2 x + 9 + 73 2 x + 9 - 73)

Add a constant to the solution

= - \frac{1}{73} (2 x + 9) (ln \frac{1}{73} (2 x + 9) + 1 - ln \frac{1}{73} (2 x + 9) - 1) + \frac{2}{73} x ln \frac{1}{73} (2 x + 9) + 1 - \frac{2}{73} x ln \frac{1}{73} (2 x + 9) - 1 + \frac{9}{73} ln \frac{1}{73} (2 x + 9) + 1 - \frac{9}{73} ln \frac{1}{73} (2 x + 9) - 1 + ln (2 x + 9 + 73 2 x + 9 - 73) + C

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What is the integral of (2x+9)/(x^2+9x+2) ?
The integral of (2x+9)/(x^2+9x+2) is