Upgrading to LaTeX For Mathematical Posts

This past week I spent roughly 10 hours learning LaTeX. I figured if I was actually going to use it to the full extent it should be merited I'd share

Hume: The Essential Philosophical Works | Part II - A Treatise of Human Nature | Book 1 - Of Understanding | Part 1 - Of Ideas, Their Origin, Composition, Connexion, Abstraction, etc. | Section 1 - Of the Origin of Our Ideas | pp. 9 - 14

Pre-Reflection First off what even is a treatise? A treatise is a systematic, written argument using methodical discussion(s) of the facts, the principles involved, and the conclusions reached. Just wanted

Integral from 0 to ∞ [x^a/(x^b+1)] dx

\begin{align*}& I(a, b) = \int_0^\infty \dfrac{x^a}{x^b+1} dx =\dfrac{\beta(\dfrac{a+1}{b}, 1-\dfrac{a+1}{b})}{b} \\& \\& {\rm{Proof}}: \\& \\& \therefore u=x^b \Longrightarrow \int_0^\infty \dfrac{u^{\frac{a-b+1}{b}}}{b(u+1)} du \\& \\& \therefore \int_0^\infty \dfrac{u^{\frac{a-b+1}{b}}}{b(u+1)} du = \dfrac{\beta(\dfrac{a+1}{b}, 1-\dfrac{a+1}{b})}{b} \\&

200m MLR Estimate

16 April 2019 Formula (19/718 sampled) \begin{align*} \rm{200m} &\approx \rm{100m} \left[ 1.997383 + \left( \dfrac{7.633557}{\rm{60m}} \right) \right] - 11.1841 \pm 0.3 \\& \approx\rm{100m} \left[ 2 + \left( \dfrac{7.65}{\rm{60m}} \right) \right]

Integral of [a(x^n)+b]/[c(x)+d]

\begin{align*}I = \displaystyle \int_\alpha^\beta\dfrac{a(x)^n+b}{cx+d} \: dx &= \displaystyle \dfrac{a}{c} \int_\alpha^\beta\dfrac{(x)^n+ \frac{b}{a}}{x+\frac{d}{c}} \: dx \\&= \displaystyle \dfrac{a}{c} \int_\alpha^\beta\dfrac{(x)^n+ \Delta_1}{x+\Delta_2} \: dx \\& \therefore \: u=x+\Delta_2 \: \Longrightarrow \: u\mid_{\alpha+\Delta_2=\Delta_3}^{\beta+\Delta_2=\Delta_4} \\& \therefore

Integral of 1/(x^n+1) from 0 to ∞

\begin{aligned} I = \int_0^\infty \dfrac{1}{(x)^n+1} dx  = \dfrac{\pi}{n\sin{( \frac{\pi}{n}} )} \end{aligned} \begin{aligned} \therefore \rho=\left[ (x)^n+1 \right]^{-1} \:;\: \rho \mid_0^1 \: \Longrightarrow d\rho=\dfrac{-n(x)^{n-1}}{\left[ (x)^n+1 \right]^2} \: dx\end{aligned} \begin{aligned} \therefore (n)^{-1} \int_0^1 \left[

Integral of 1/[(x^3)+1] from 0 to ∞

I=∫∞01x3+1dx=2π31.5≈1.209199576156145 Partial Fraction Decomposition Method \begin{aligned} \int_0^{\infty} \dfrac{1}{(x+1)(x^2-x+1)} \,dx \end{aligned}.\begin{aligned} \int_0^{\infty} \dfrac{1}{3(x+1)} \,dx+ \int_0^{\infty} \dfrac{x-2}{3(x^2-x+1)} \,dx \end{aligned}.\begin{aligned} \dfrac{1}{3} \int_0^{\infty} \dfrac{1}{x+1} \,dx-\dfrac{1}{3} \int_0^{\infty} \dfrac{x-2}{x^2-x+1} \,dx \end{aligned}.\begin{aligned} \left[\dfrac{1}{3} \ln{(x+1)}\right]\mid_0^{\infty}-\dfrac{1}{6}\int_0^{\infty} \dfrac{2x-1}{x^2-x+1} \,dx+\dfrac{1}{2}\int_0^{\infty} \dfrac{1}{x^2-x+1}