What is the integral of 1/(5x+x^2) ?

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

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integral of 1/(5x+x^2)

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

\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \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

= \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 ∣

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

Add a constant to the solution

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

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