What is the integral of (x+1)/(x+9) ?

The integral of (x+1)/(x+9) is -8ln|x+9|+x+C

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integral of (x+1)/(x+9)

\int \frac{x + 1}{x + 9} d x

- 8 ln ∣ x + 9 ∣ + x + C

Solution steps

\int \frac{x + 1}{x + 9} d x

\frac{x + 1}{x + 9} = - \frac{8}{x + 9} + 1

= \int - \frac{8}{x + 9} + 1 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{8}{x + 9} d x + \int 1 d x

\int \frac{8}{x + 9} d x = 8 ln ∣ x + 9 ∣

\int 1 d x = x

= - 8 ln ∣ x + 9 ∣ + x

Add a constant to the solution

= - 8 ln ∣ x + 9 ∣ + x + C

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What is the integral of (x+1)/(x+9) ?
The integral of (x+1)/(x+9) is -8ln|x+9|+x+C