What is the integral of (9x+25)/(x^2+10x+9) ?

The integral of (9x+25)/(x^2+10x+9) is 2ln|x+1|+7ln|x+9|+C

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integral of (9x+25)/(x^2+10x+9)

\int \frac{9 x + 25}{x ^{2} + 10 x + 9} d x

2 ln ∣ x + 1 ∣ + 7 ln ∣ x + 9 ∣ + C

Solution steps

\int \frac{9 x + 25}{x ^{2} + 10 x + 9} d x

Take the partial fraction of

\frac{9 x + 25}{x ^{2} + 10 x + 9} : \frac{2}{x + 1} + \frac{7}{x + 9}

= \int \frac{2}{x + 1} + \frac{7}{x + 9} 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{2}{x + 1} d x + \int \frac{7}{x + 9} d x

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

\int \frac{7}{x + 9} d x = 7 ln ∣ x + 9 ∣

= 2 ln ∣ x + 1 ∣ + 7 ln ∣ x + 9 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + 7 ln ∣ x + 9 ∣ + C

What is the integral of (9x+25)/(x^2+10x+9) ?
The integral of (9x+25)/(x^2+10x+9) is 2ln|x+1|+7ln|x+9|+C