Derivatives #

$$\begin{align*} \frac{d}{dx} \tan(x) &= 1 + \tan^2(x) = \sec^2(x) \\ \frac{d}{dx} \csc(x) &= -\cot(x) \cdot \csc(x) \\ \frac{d}{dx} \sec(x) &= \frac{\sin(x)}{\cos^2(x)} = \tan(x) \cdot \sec(x) \\ \frac{d}{dx} \cot(x) &= -\csc^2(x) \\ \frac{d}{dx} \log_a(x) &= \frac{1}{x\cdot \ln(a)} \\ \frac{d}{dx} a^u &= a^x \cdot \ln (a) du \\ \frac{d}{dx} \sin^2(x) &= \sin(2x) \\ \frac{d}{dx} \cos^2(x) &= -\sin(2x) \\ \frac{d}{dx} \tan^2(x) &= 2\tan(x)\cdot \sec^2(x) \end{align*}$$

Integrals #

$$\begin{align*} \int a^x dx &= \bigg(\frac{1}{\ln(a)}\bigg) a^x + C \\ \int \tan(x) dx &= -\ln\vert \cos(x) \vert + C \\ \int \tan^2(x) dx &= \tan(x) - x + C \\ \int \csc(x) dx &= \ln\vert \csc(x) - \cot(x)\vert + C = \ln \bigg\vert \tan\bigg(\frac{x}{2}\bigg)\bigg\vert + C \\ \int \csc^2(x) dx &= -\cot(x) + C \\ \int \sec(x) dx &= -\ln\vert \sec(x) + \tan(x)\vert + C \\ \int \sec^2(x) dx &= \tan(x) + C \\ \int \cot(x) dx &= \ln\vert \sin(x) \vert + C \\ \int \cot^2(x) dx &= -\cot(x) - x + C \\ \int \frac{1}{\sin(ax)\cos(ax)} &= \frac{1}{a} \ln\vert\tan(ax)\vert + C \\ \int \frac{1}{x\sqrt{x^2-a^2}} dx &= \frac{1}{a} \sec^{-1}\bigg( \frac{\vert x \vert}{a}\bigg) + C \\ \int \frac{1}{\sqrt{a^2-x^2}} dx &= \sin^{-1}\bigg( \frac{x}{a} \bigg) + C \\ \int \frac{1}{a^2 + x^2} dx &= \frac{1}{a} \tan^{-1}\bigg( \frac{x}{a} \bigg) + C \end{align*}$$

Many more integrals

Triangle Sub #

$$\begin{align*} \sqrt{b^2x^2-a^2} &\Longrightarrow x = \frac{a}{b}\cdot\sec\theta; \theta \in [0, \pi/2), (\pi/2, \pi] \\ \sqrt{a^2-b^2x^2} &\Longrightarrow x = \frac{a}{b}\cdot\sin\theta; \theta \in [-\pi/2, \pi/2] \\ \sqrt{a^2+b^2x^2} &\Longrightarrow x = \frac{a}{b}\cdot\tan\theta; \theta \in (-\pi/2, \pi/2) \end{align*}$$