What is the integral of (5x+3)/(x^2+x) ?

The integral of (5x+3)/(x^2+x) is 3ln|x|+2ln|x+1|+C

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integral of (5x+3)/(x^2+x)

\int \frac{5 x + 3}{x ^{2} + x} d x

3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{5 x + 3}{x ^{2} + x} d x

Take the partial fraction of

\frac{5 x + 3}{x ^{2} + x} : \frac{3}{x} + \frac{2}{x + 1}

= \int \frac{3}{x} + \frac{2}{x + 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{3}{x} d x + \int \frac{2}{x + 1} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= 3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣

Add a constant to the solution

= 3 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ + C

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What is the integral of (5x+3)/(x^2+x) ?
The integral of (5x+3)/(x^2+x) is 3ln|x|+2ln|x+1|+C