The Real Arctangent Function

Theorem: Injectivity of the Real Tangent Function

The restriction of the Real Tangent Function on the interval $\left(-\frac{\pi }{2};\frac{\pi }{2}\right)$ and thus admits an inverse function on it.

PROOF

TODO

Definition: Real Arctangent Function

The real arctangent function is the inverse function of the restriction of the Real Tangent Function on the interval $\left(-\frac{\pi }{2};\frac{\pi }{2}\right)$.

NOTATION

$$\arctan {\tan }^{-1}$$

Note: Domain of the Real Arctangent Function

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

Note: Domain of the Real Arctangent Function

The image of the Real Arctangent Function is the interval $\left(-\frac{\pi }{2};\frac{\pi }{2}\right)$.

Properties

Theorem: Continuity of the Real Arctangent Function

The Real Arctangent Function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Arctangent Function

The Real Arctangent Function is differentiable with

$$(\arctan x{)}^{\prime }=\frac{1}{x^2+1}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Arctangent Function

The Antiderivatives of the Real Arctangent Function are given by

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

PROOF

TODO