What is the integral of (x+4)/(x^2+x) ?

The integral of (x+4)/(x^2+x) is 4ln|x|-3ln|x+1|+C

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integral of (x+4)/(x^2+x)

\int \frac{x + 4}{x ^{2} + x} d x

4 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{x + 4}{x ^{2} + x} d x

Take the partial fraction of

\frac{x + 4}{x ^{2} + x} : \frac{4}{x} - \frac{3}{x + 1}

= \int \frac{4}{x} - \frac{3}{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{4}{x} d x - \int \frac{3}{x + 1} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

= 4 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣

Add a constant to the solution

= 4 ln ∣ x ∣ - 3 ln ∣ x + 1 ∣ + C

What is the integral of (x+4)/(x^2+x) ?
The integral of (x+4)/(x^2+x) is 4ln|x|-3ln|x+1|+C