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

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

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

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

\frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C

Solution steps

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

Expand

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

\int \frac{x}{x ^{2} + 5 x} d x = ln ∣ x + 5 ∣

\int \frac{2}{x ^{2} + 5 x} d x = 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

= ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

Simplify

ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣) : \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

Add a constant to the solution

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C

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