Calculus
Table of Contents
1. Single variable calculus
1.1. Limit of a function
\begin{equation}
\forall\varepsilon>0, \exists\delta>0, \forall x\in\mathbb{R} : 0 < |x - p| < \delta \Rightarrow |f(x) - L| < \varepsilon
\end{equation}
write
\begin{equation} \lim_{x \rightarrow p} f(x) = L \end{equation}1.1.1. One-sided limit
\begin{equation}
\forall\varepsilon>0, \exists\delta>0, \forall x\in(a,b) : 0 < x - p < \delta \Rightarrow |f(x) - L| < \varepsilon
\end{equation}
write
\begin{equation} \lim_{x \rightarrow p^+} f(x) = L \end{equation}
\begin{equation}
\forall\varepsilon>0, \exists\delta>0, \forall x\in(a,b) : 0 < p - x < \delta \Rightarrow |f(x) - L| < \varepsilon
\end{equation}
write
\begin{equation} \lim_{x \rightarrow p^-} f(x) = L \end{equation}1.2. Continuity of a function
\begin{equation}
\lim_{x \rightarrow c} f(x) = f(c)
\end{equation}
1.2.1. Continuity of a piecewise function
1.2.2. Existance of maximum and minimum
1.2.3. Intermediate value theorem (Bolzano-Cauchy theorem)
If \(u\) is a number between \(f(a)\) and \(f(b)\), that is
\begin{equation} \min(f(a), f(b)) < u < \max(f(a), f(b)) \end{equation}then there is a \(c \in (a,b)\) such that \(f(c) = u\).
1.3. Differentiation
1.4. Integration
1.4.1. Integrals of simple functions
- Rational functions
- \(\displaystyle \int a\mathrm{d}x = ax + C\)
Functions with singularity at \(0\) for \(n \le -1\)
- \(\displaystyle \int x^n\mathrm{d}x = \frac{x^{n+1}}{n+1} + C\) (\(n \neq -1\))
- \(\displaystyle \int \frac{1}{x}\mathrm{d}x = \ln|x| + C\)
- Exponential functions
- \(\displaystyle \int e^x\mathrm{d}x = e^x + C\)
- Logarithms
- \(\displaystyle \int\ln x\mathrm{d}x = x (\ln x - 1)\)
- Trigonometric functions
- \(\displaystyle \int\sin x\mathrm{d}x = -\cos x + C\)
- \(\displaystyle \int\cos x\mathrm{d}x = \sin x + C\)
- \(\displaystyle \int\tan x\mathrm{d}x = -\ln|\cos x| + C\)
- \(\displaystyle \int\cot x\mathrm{d}x = \ln|\sin x| + C\)
- Inverse trigonometric functions
- \(\displaystyle \int\arcsin x\mathrm{d}x = x\arcsin x + \sqrt{1 - x^2} + C\), for \(|x|\le1\)
- \(\displaystyle \int\arccos x\mathrm{d}x = x\arccos x - \sqrt{1 - x^2} + C\), for \(|x|\le1\)
- \(\displaystyle \int\arctan x\mathrm{d}x = x\arctan x - \frac{1}{2}\ln|1 + x^2| + C\), for all \(x\in\mathbb{R}\)
- \(\displaystyle \int\arccot x\mathrm{d}x = x\arccot x + \frac{1}{2}\ln|1 + x^2| + C\), for all \(x\in\mathbb{R}\)
1.4.2. Integrals derived from integrals of simple functions
- Rational functions
Functions with singularity at \(0\) for \(n \le -1\)
- \(\displaystyle \int (ax+b)^n\mathrm{d}x = \frac{(ax+b)^{n+1}}{a(n+1)} + C\) (\(n \neq -1\))
- \(\displaystyle \int \frac{c}{ax+b}\mathrm{d}x = \frac{c}{a}\ln|ax+b| + C\)
- Exponential functions
- \(\displaystyle \int e^{ax}\mathrm{d}x = \frac{1}{a}e^{ax} + C\)
- \(\displaystyle \int f'(x)e^{f(x)}\mathrm{d}x = e^{f(x)} + C\)
- \(\displaystyle \int a^x\mathrm{d}x = \frac{a^x}{\ln a} + C\)
- \(\displaystyle \int e^x\bigl[f(x) + f'(x)\bigr]\mathrm{d}x = e^x f(x) + C\)
- \(\displaystyle \int e^x \biggl[f(x) - (-1)^n \frac{\mathrm{d}^n f(x)}{\mathrm{d}x^n}\biggr]\mathrm{d}x = e^x \sum_{k=1}^n (-1)^{k-1} \frac{\mathrm{d}^{k-1} f(x)}{\mathrm{d}x^{k-1}} + C\)
- \(\displaystyle \forall n\in\mathbb{N}^{*} \biggl[f(x) - \frac{\mathrm{d}^n f(x)}{\mathrm{d}x^n}\biggr]\mathrm{d}x = -e^{-x} \sum_{k=1}^n \frac{\mathrm{d}^{k-1} f(x)}{\mathrm{d}x^{k-1}} + C\)
- Product of \(e^{cx}\) and \(x^n\)
- \(\displaystyle \int xe^{cx}\mathrm{d}x = e^{cx} \biggl(\frac{cx-1}{c^2}\biggr)\), for \(c \neq 0\)
- \(\displaystyle \int x^2 e^{cx}\mathrm{d}x = e^{cx} \biggl(\frac{x^2}{c} - 2 \frac{cx - 1}{c^2}\biggr)\)
- \(\displaystyle \int x^n e^{cx}\mathrm{d}x = \frac{1}{c}x^n e^{cx} - \frac{n}{c} \int x^{n-1}e^{cx}\mathrm{x}\)
- Logarithms
- \(\displaystyle \int\log_a x\mathrm{d}x = \frac{x}{\ln a} (\ln x - 1) + C\)
- \(\displaystyle \int\ln(ax)\mathrm{d}x = x (\ln(ax) - 1)\)
- \(\displaystyle \int\ln(ax+b)\mathrm{d}x = \frac{ax+b}{a} (\ln(ax+b) - 1)\)
- \(\displaystyle \int\ln^2 x\mathrm{d}x = x \ln^2 x - 2x\ln x + 2x\)
- Product of \(\ln x\) and \(x^n\)
- \(\displaystyle \int x^n \ln x\mathrm{d}x = x^{n+1} \biggl(\frac{\ln x}{n+1} - \frac{1}{(n+1)^2}\biggr)\) for \(n \neq -1\)
- \(\displaystyle \int x^n \ln^m x\mathrm{d}x = \frac{x^{n+1}\ln^m x}{n+1} - \frac{m}{n+1} \int x^n \ln^{m-1}x\mathrm{d}x\) for \(m \neq -1\)
- \(\displaystyle \int \frac{\ln^m x}{x}\mathrm{d}x = \frac{\ln^{m+1}x}{m+1}\) for \(m \neq -1\)
- \(\displaystyle \int \frac{\ln x}{x^n}\mathrm{d}x = -\frac{\ln x}{(n-1)x^{n-1}} - \frac{1}{(n-1)^2 x^{n-1}}\) for \(n \neq -1\)
- \(\displaystyle \int \frac{\ln^m x}{x^n}\mathrm{d}x = -\frac{\ln^m x}{(n-1)x^{n-1}} + \frac{m}{n-1} \int \frac{\ln^{m-1}x}{x^n}\mathrm{d}x\) for \(n \neq -1\)
- \(\displaystyle \int \frac{x^n}{\ln^m x}\mathrm{d}x = -\frac{x^{n+1}}{(m-1) \ln^{m-1}x} + \frac{n+1}{m-1} \int \frac{x^n}{\ln^{m-1}x}\mathrm{d}x\) for \(m \neq -1\)
- \(\displaystyle \int \frac{1}{x\ln x}\mathrm{d}x = \ln|\ln x|\)
- \(\displaystyle \int \frac{1}{x\ln x \ln \ln x}\mathrm{d}x = \ln|\ln|\ln x||\)
- \(\displaystyle \int \frac{1}{x(\ln x)^n}\mathrm{d}x = -\frac{1}{(n-1)(\ln x)^{n-1}}\) for \(n \neq 1\)
- \(\displaystyle \ln(x^2 + a^2) \mathrm{d}x = x\ln(x^2 + a^2) - 2x + 2a\tan^{-1}\frac{x}{a}\)
- \(\displaystyle \frac{x}{x^2 + a^2}\ln(x^2 + a^2) \mathrm{d}x = \frac{1}{4}\ln^2(x^2 + a^2)\)
- Trigonometric functions
- \(\displaystyle \int\sin^2 x\mathrm{d}x = \frac{1}{2} \biggl(x - \frac{\sin 2x}{2}\biggr) + C = \frac{1}{2} \biggl(x - \sin x \cos x\biggr) + C\)
- \(\displaystyle \int\cos^2 x\mathrm{d}x = \frac{1}{2} \biggl(x + \frac{\sin 2x}{2}\biggr) + C = \frac{1}{2} \biggl(x + \sin x \cos x\biggr) + C\)
- \(\displaystyle \int\tan^2 x\mathrm{d}x = \tan x - x + C\)
- \(\displaystyle \int\cot^2 x\mathrm{d}x = -\cot x - x + C\)