Primitive of Arccotangent Function

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Theorem

$\ds \int \arccot x \rd x = x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C$


Proof

From Primitive of $\arccot \dfrac x a$:

$\ds \int \arctan \frac x a \rd x = x \arccot \frac x a + \frac a 2 \map \ln {x^2 + a^2} + C$

The result follows by setting $a = 1$.

$\blacksquare$


Also presented as

This result can also be presented as:

$\ds \int \arccot x \rd x = x \arccot x + \ln \sqrt {x^2 + 1} + C$


Also see


Sources