Single variable calculus
1.	Limits of sequences
2.	Limits of functions. Continuity
3.	Derivatives
4.	Basic properties of derivatives
5.	Chain rule
6.	Maximum and minimum of a function
7.	L'Hopital's Theorem
8.	Antiderivatives and indefinite integrals
9.	Definite integrals. Fundamental theorem of calculus
10.	Substitution
11.	Integration by parts
12.	Integration of rational functions
13.	Integration of trigonometric functions
14.	Trigonometric substitutions
	Multivariable calculus
15.	Vectors. Equations of lines and planes
16.	Functions of several variables
17.	Vector fields. Parametric equations of curves and surfaces
18.	Partial derivatives
19.	Tangent planes
20.	Gradient vector
21.	Chain rule
22.	Lagrange multipliers
23.	Double integrals
24.	Triple integrals
25.	Change of variables in multiple integrals
26.	Line integrals
27.	Surface integrals
28.	Curl and divergence
29.	Green’s theorem, divergence theorem, and Stokes’ theorem
30.	Formulas and summary

Formulas in Calculus and Pre-calculus

Trigonometry and pre-calculus

\(\displaystyle (a+b)^2=a^2+2ab+b^2\) \( \displaystyle (a+b)(c+d)=ac+ad+bc+bd\) \(\displaystyle n!=1\cdot2\cdots\cdot (n-1)\cdot n\) \(\displaystyle \binom{n}k=\frac{n!}{k!(n-k)!}\) \(\displaystyle (a+b)^n=\sum_{i=0}^n\binom{n}ia^ib^{n-i}\) \(\displaystyle a^2-b^2=(a+b)(a-b)\) \(\displaystyle (ab)^n=a^nb^n\) \(\displaystyle \left(a^x\right)^y=a^{xy}\) \(\displaystyle e^{\ln x}=x\) \(\displaystyle a^x=e^{x\ln a}\) \(\displaystyle \ln(ab)=\ln a+\ln b\) \(\displaystyle \ln\left(a^k\right)=k\ln a\)

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\(\displaystyle \sin\theta=\frac{a}{c}\) \(\displaystyle \cos\theta=\frac{b}{c}\)

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\(\displaystyle \sin^2\alpha+\cos^2\alpha=1\) \(\displaystyle \tan\alpha=\frac{\sin\alpha}{\cos\alpha}\) \(\displaystyle \cot\alpha=\frac{\cos\alpha}{\sin\alpha}\) \(\displaystyle \sec \alpha=\frac1{\cos \alpha}\) \(\displaystyle \sec^2\alpha=1+\tan^2\alpha\) \(\displaystyle \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\) \(\displaystyle \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\) \(\displaystyle \cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2\) \(\displaystyle \cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2\) \(\displaystyle \sin\alpha+\sin\beta=2\cos\frac{\alpha-\beta}2\sin\frac{\alpha+\beta}2\) \(\displaystyle \sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}2\sin\frac{\alpha-\beta}2\) \(\displaystyle \cos(2\alpha)=\cos^2\alpha-\sin^2\alpha\) \(\displaystyle \sin(2\alpha)=2\sin\alpha\cos\alpha\) \(\displaystyle \cos^2\alpha=\frac{1+\cos(2\alpha)}2\) \(\displaystyle \sin^2\alpha=\frac{1-\cos(2\alpha)}2\)

Derivatives

\(\displaystyle \frac{d}{dx} C=0\) \(\displaystyle \frac{d}{dx} x=1\) \(\displaystyle \frac{d}{dx} x^k=kx^{k-1}\), for \(k\in\mathbb R\) \(\displaystyle \frac{d}{dx} \ln x=\frac1x\) \(\displaystyle \frac{d}{dx} e^x=e^x\) \(\displaystyle \frac{d}{dx} a^x=a^x\ln a\), for \(a > 0\)

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\(\displaystyle \frac{d}{dx}\sin x=\cos x\) \(\displaystyle\frac{d}{dx}\cos x=-\sin x\) \(\displaystyle \frac{d}{dx} \tan x=\frac1{\cos^2x}\) \(\displaystyle \frac{d}{dx} \cot x=-\frac1{\sin^2x}\) \(\displaystyle \frac{d}{dx} \arcsin x=\frac1{\sqrt{1-x^2}}\) \(\displaystyle \frac{d}{dx}\arccos x=-\frac1{\sqrt{1-x^2}}\)

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\(\displaystyle \frac{d}{dx}\arctan x=\frac1{1+x^2}\) \(\displaystyle \frac{d}{dx}\sinh x=\cosh x\) \(\displaystyle \frac{d}{dx}\cosh x=\sinh x\) \(\displaystyle \frac{d}{dx}\tanh x=\frac1{\cosh^2 x}\) \(\displaystyle \frac{d}{dx}\coth x=-\frac1{\sinh^2 x}\) \(\displaystyle \left(f(x)\cdot g(x)\right)^{\prime}=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)\) \(\displaystyle \left(\frac{f(x)}{g(x)}\right)^{\prime}=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{\left(g(x)\right)^2}\) \(\displaystyle \left( f(g(x))\right)^{\prime}=f^{\prime}(g(x))\cdot g^{\prime}(x)\)

Integrals

\(\displaystyle \int x^p\,dx= \frac{x^{p+1}}{p+1}+C\), for \(p\neq -1\) \(\displaystyle \int \frac1x\,dx= \ln\left|x\right|+C\) \(\displaystyle\int e^x\,dx= e^x+C\) \(\displaystyle\int a^x\,dx= \frac{a^x}{\ln a}+C\), for \(a > 0\) and \(a\neq 1\) \(\displaystyle\int \ln x\,dx=x\ln x-x+C\) \(\displaystyle\int \cos x\,dx=\sin x+C\) \(\displaystyle\int \sin x\,dx=-\cos x+C\) \(\displaystyle \int \frac1{\cos^2x}\,dx=\tan x+C\) \(\displaystyle \int \frac1{\sin^2x}\,dx=-\cot x+C\) \(\displaystyle \int \sec x\cdot \tan x\,dx=\sec x+C\) \(\displaystyle \int\tan x\,dx=-\ln\left|\cos x\right|+C\) \(\displaystyle \int\cot x\,dx=\ln\left|\sin x\right|+C\) \(\displaystyle \int \arcsin x\,dx=x\arcsin x+\sqrt{1-x^2}+C\) \(\displaystyle \int \arctan x\,dx=x\arctan x-\frac12\ln\left(1+x^2\right)+C\)

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\(\displaystyle \int e^{ax}\sin(bx)\,dx =\frac{e^{ax}\left(a\sin(bx)-b\cos(bx)\right)}{a^2+b^2}+C\) \(\displaystyle \int e^{ax}\cos(bx)\,dx=\frac{e^{ax}\left(a\cos(bx)+b\sin(bx)\right)}{a^2+b^2}+C\) \(\displaystyle \int \frac{dx}{x^2+a^2}=\frac1a\arctan\frac{x}{a}+C\) \(\displaystyle \int \frac{dx}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C\) \(\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{a}+C\) \(\displaystyle \int\frac{dx}{\sqrt{x^2\pm a^2}}=\ln\left|x+\sqrt{x^2\pm a^2}\right|+C\) \(\displaystyle \int\frac{dx}{\sqrt{x^2\pm a^2}^3}=\pm\frac{x}{a^2\sqrt{x^2\pm a^2}}+C\) \(\displaystyle \int \sqrt{x^2\pm a^2}\,dx=\pm\frac{a^2}2\ln\left|x +\sqrt{x^2\pm a^2}\right|\) \(\displaystyle+\frac{x}2\sqrt{x^2\pm a^2}+C\) \(\displaystyle \int \sqrt{x^2\pm a^2}^3\,dx=\frac{x}8\left(2x^2\pm5a^2\right)\sqrt{x^2\pm a^2}\) \(\displaystyle+\frac{3a^4}8\ln\left|x+\sqrt{x^2\pm a^2}\right|+ C\) \(\displaystyle \int_{-\infty}^{+\infty}e^{-x^2}\,dx=\sqrt{\pi}\) \(\displaystyle \int_0^{+\infty}\frac{\sin x}x\,dx=\frac{\pi}2\) \(\displaystyle \int_0^{\frac{\pi}2}\sin^{2n}x\,dx=\frac{(2n-1)!!}{(2n)!!}\cdot\frac{\pi}2\), for \(n\in\mathbb N\cup\{0\}\) \(\displaystyle \int_0^{\frac{\pi}2}\sin^{2n+1}x\,dx=\frac{(2n)!!}{(2n+1)!!}\), for \(n\in\mathbb N\cup \{0\}\)