User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

Theorem

 * $\displaystyle \int \csc x \ \mathrm d x = -\ln \left \vert {\csc x + \cot x} \right \vert + C$

where $\csc x + \cot x \ne 0$.

Proof
We make the Weierstrass Substitution:


 * $\displaystyle u = \tan \frac x 2$

for $x \in \left\{ {-\pi < x < \pi: \cos x \ne 0} \right \}$

That is:

Proof
Let $u = x + L$.

Then $\displaystyle \frac {\mathrm du}{\mathrm dx} = 1 + 0$

So:

When $C = 0$, $f$ is clearly periodic.

For all other $C$, we invoke Periodic Function plus Constant.