#+title:Table of Integrals
#+setupfile: ../../math_options.org

* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)

* $a + bx$
\begin{align*}
\intg{(a + bx)^n}{\frac{(a+bx)^{n+1}}{b(n+1)}}\\
\int\frac{\D{x}}{a + bx} &= \frac{1}{b}\log(a+bx) + C\\
\int\frac{x\D{x}}{a + bx} &= \frac{1}{b^2}(a + bx - a\log(a+bx)) + C
\end{align*}

* Trigonometric Funtions
\begin{align*}
\intg{\sin(x)}{-\cos(x)}\\
\intg{\cos(x)}{\sin(x)}\\
\intg{\tan(x)}{-\ln(\cos(x))}\\
\intg{\sec(x)}{\ln(\sec(x) + \tan(x))}\\
\intg{\csc(x)}{-\ln(\csc(x) + \cot(x))}\\
\intg{\cot(x)}{\ln(\sin(x))}
\end{align*}
* Hyperbolic Funtions as results
\begin{align*}
\int\frac{\text{d}x}{\sqrt{x^2 + a^2}} &= \arsinh{\frac{x}{a}} + C\\
\int\frac{\text{d}x}{\sqrt{x^2 - a^2}} &= \arcosh{\frac{x}{a}} + C\\
\int\frac{\text{d}x}{a^2 - x^2} &= \frac{1}{a}\artanh{\frac{x}{a}} + C\\
%\int\frac{\text{d}x}{x^2 - a^2} &= -\frac{1}{a}\arcoth{\frac{x}{a}} + C\\
\int\frac{\text{d}x}{x\sqrt{a^2 - x^2}} &= -\frac{1}{a}\arsech{\frac{x}{a}} + C\\
\int\frac{\text{d}x}{x\sqrt{x^2 + a^2}} &= -\frac{1}{a}\arcsch{\frac{x}{a}} + C\\
\intg{\sqrt{x^2+a^2}}{\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\arsinh{\frac{x}{a}}}\\
\intg{\sqrt{x^2-a^2}}{\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\arcosh{\frac{x}{a}}}
\end{align*}