What is the integral of (1/x-5/(9x^2+1)) ?

The integral of (1/x-5/(9x^2+1)) is ln|x|-5/3 arctan(3x)+C

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integral of (1/x-5/(9x^2+1))

\int (\frac{1}{x} - \frac{5}{9 x ^{2} + 1}) d x

ln ∣ x ∣ - \frac{5}{3} arctan (3 x) + C

Solution steps

\int \frac{1}{x} - \frac{5}{9 x ^{2} + 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{1}{x} d x - \int \frac{5}{9 x ^{2} + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{5}{9 x ^{2} + 1} d x = \frac{5}{3} arctan (3 x)

= ln ∣ x ∣ - \frac{5}{3} arctan (3 x)

Add a constant to the solution

= ln ∣ x ∣ - \frac{5}{3} arctan (3 x) + C

What is the integral of (1/x-5/(9x^2+1)) ?
The integral of (1/x-5/(9x^2+1)) is ln|x|-5/3 arctan(3x)+C