What is the integral of (arccot(3x))/(1+9x^2) ?

The integral of (arccot(3x))/(1+9x^2) is 1/3 arccot(3x)arctan(3x)+1/6 arctan^2(3x)+C

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integral of (arccot(3x))/(1+9x^2)

\int \frac{arccot ( 3 x )}{1 + 9 x ^{2}} d x

\frac{1}{3} arccot (3 x) arctan (3 x) + \frac{1}{6} arctan^{2} (3 x) + C

Solution steps

\int \frac{arccot ( 3 x )}{1 + 9 x ^{2}} d x

Apply Integration By Parts

: \frac{1}{3} arccot (3 x) arctan (3 x) - \int - \frac{arctan ( 3 x )}{9 x ^{2} + 1} d x

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

\int - \frac{arctan ( 3 x )}{9 x ^{2} + 1} d x = - \frac{1}{6} arctan^{2} (3 x)

= \frac{1}{3} arccot (3 x) arctan (3 x) - (- \frac{1}{6} arctan^{2} (3 x))

Simplify

= \frac{1}{3} arccot (3 x) arctan (3 x) + \frac{1}{6} arctan^{2} (3 x)

Add a constant to the solution

= \frac{1}{3} arccot (3 x) arctan (3 x) + \frac{1}{6} arctan^{2} (3 x) + C

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What is the integral of (arccot(3x))/(1+9x^2) ?
The integral of (arccot(3x))/(1+9x^2) is 1/3 arccot(3x)arctan(3x)+1/6 arctan^2(3x)+C