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

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

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

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

ln ∣ x ∣ - ln ∣ x + 5 ∣ + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{x ^{2} + 5 x} d x

Take the partial fraction of

\frac{1}{x ^{2} + 5 x} : \frac{1}{5 x} - \frac{1}{5 ( x + 5 )}

= 5 \cdot \int \frac{1}{5 x} - \frac{1}{5 ( x + 5 )} 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

= 5 (\int \frac{1}{5 x} d x - \int \frac{1}{5 ( x + 5 )} d x)

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

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

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

Simplify

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

= ln ∣ x ∣ - ln ∣ x + 5 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x + 5 ∣ + C

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