The Real Arccotangent Function

Theorem: Injectivity of the Real Cotangent Function

The restriction of the Real Cotangent Function on the interval $(0;\pi )$ and thus admits an inverse function on it.

PROOF

TODO

Definition: Real Arccotangent Function

The real arccotangent function is the inverse function of the restriction of the Real Cotangent Function on the interval $(0;\pi )$.

NOTATION

$$\mathrm{arccot}{\cot }^{-1}$$

Note: Domain of the Real Arctangent Function

The domain of the Real Cotangent Function is $\mathbb{R}$.

Note: Domain of the Real Arctangent Function

The image of the Real Cotangent Function is the interval $(0;\pi )$.

Properties

Theorem: Continuity of the Real Arccotangent Function

The Real Arccotangent Function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Arccotangent Function

The Real Arccotangent Function is differentiable with

$$(\mathrm{arccot}x{)}^{\prime }=-\frac{1}{x^2+1}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Arccotangent Function

The Antiderivatives of the Real Arccotangent Function are given by

$$\int \mathrm{arccot}(x)\mathrm{dx}=x\mathrm{arccot}(x)+\frac{1}{2}\ln (1+x^2)+\text{const}$$

PROOF

TODO