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Random math formulas

$$\frac1{1-x}=1+x+x^2+x^3+x^4+\cdots=\sum_{k=0}^{\infty}x^k,\,\, \vert x\vert <1$$

$$\int_0^{\infty}e^{-2x}\, \mathrm dx= \lim_{b \to \infty}\int_0^be^{-2x}\, \mathrm dx = \frac 12$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

$$\int\frac1u\,\mathrm du=\ln\vert u \vert + C$$

$$\frac{\mathrm d}{\mathrm dx}[\sec^{-1}x]=\frac1{\vert x\vert\sqrt{x^2-1}}$$

$$\lim_{x\to 0}\frac{\sin x}x=1$$