What is the integral of arctan(x/3) ?

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

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integral of arctan(x/3)

\int arctan (\frac{x}{3}) d x

x arctan (\frac{x}{3}) - \frac{3}{2} ln x^{2} + 9 + C

Solution steps

\int arctan (\frac{x}{3}) d x

Apply Integration By Parts

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

\int \frac{3 x}{x ^{2} + 9} d x = \frac{3}{2} ln x^{2} + 9

= x arctan (\frac{x}{3}) - \frac{3}{2} ln x^{2} + 9

Add a constant to the solution

= x arctan (\frac{x}{3}) - \frac{3}{2} ln x^{2} + 9 + C

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