/ math / analysis /

# Integral 

##### Definition

An integral is a function $F$ of which a given function $f$ is the derivative. The integral describes displacement, area, volume, and other concepts that arise by calculating the infinite sum of rectangles of infinitesimal width. The symbol ${\,\text{d}}x$ was taken to represent an infinitesimally "small piece" of the independent variable $x$.

Indefinite integral:

$$F(x)=\int f(x){\,\text{d}}x$$

with the function $f$ of a real variable $x$.

Definite integral:

$$\int_{a}^{b}\,f(x){\,\text{d}}x = \left[F(x)\right]_{a}^{b}=F(b)-F(a)$$

with the interval $[a, b]$ of the real line.

##### Example

Given $f(x) = x^2$, then $\int_1^2 x^2 {\,\text{d}}x = \left[ \frac{1}{3}x^3 \right]_1^2 = \frac{8}{3} - \frac{1}{3} = \frac{7}{3} = 2 \frac{1}{3}$

## Common Integrals

$F(x) - C$ $f(x)$ $f'(x)$
$\frac{1}{q+1}x^{q+1}$ $x^q$ $qx^{q-1}$
$\frac{2\sqrt{ax^3}}{3}$ $\sqrt{ax}$ $\frac{a}{2\sqrt{ax}}$
$x\ln(ax) -x$ $\ln(ax)$ $\textstyle \frac{1}{x}$
$\frac{1}{a^2} e^{ax}(ax- 1)$ $x \cdot e^{ax}$ $e^{ax}(ax+1)$
$\frac{a^x}{\ln(a)}$ $a^x$ $a^x \ln(a)$
$-\cos(x)$ $\sin(x)$ $\cos(x)$
$\sin(x)$ $\cos(x)$ $-\sin(x)$
$-\ln {\left\vert{\cos(x)}\right\vert}$ $\tan(x)$ $\frac{1}{\cos^2(x)}$
$\ln {\left\vert{\sin(x)}\right\vert}$ $\cot(x)$ $\frac{-1}{\sin^2(x)}$
$x\arcsin (x)+\sqrt{1-x^2}$ $\arcsin(x)$ $\frac{1}{\sqrt{1-x^2}}$
$x\arccos (x)-\sqrt{1-x^2}$ $\arccos(x)$ $-\frac{1}{\sqrt{1-x^2}}$
$x\arctan (x)-\frac{1}{2} \ln {\left\vert{1+ x^2}\right\vert}$ $\arctan(x)$ $\frac{1}{1+x^2}$
$x\arctan (x)+\frac{1}{2} \ln {\left\vert{1+ x^2}\right\vert}$ ${\text{arccot}}(x)$ $-\frac{1}{1+x^2}$
$\cosh(x)$ $\sinh(x)$ $\cosh (x)$
$\sinh(x)$ $\cosh(x)$ $\sinh (x)$

## Trigonometry

$\int x \cos(x) {\,\text{d}}x = \cos(x) + x \sin(x)$
$\int x \sin(x) {\,\text{d}}x = \sin(x) - x \cos(x)$
$\int \sin^2(x) {\,\text{d}}x = \frac12 \bigl(x - \sin(x)\cos(x) \bigr)$
$\int \cos^2(x) {\,\text{d}}x = \frac12 \bigl(x + \sin(x)\cos(x) \bigr)$
$\int \cos(x)\sin(x) = -\frac12 \cos^2(x)$

## Line Integrals

Given a curve ${\boldsymbol \gamma}:[a,b] \rightarrow {\mathop{\mathbb R}}^n, t \mapsto {\boldsymbol \gamma}(t)$.

$$\int\limits_\gamma f {\,\text{d}}s := \int\limits^b_a f\bigl({\boldsymbol \gamma(t)}\bigr) \cdot {\left\lVert{{\boldsymbol {\dot{\gamma}} }(t)}\right\rVert} {\,\text{d}}t$$

with the scalar field $f({\boldsymbol x})$ along a curve ${\boldsymbol \gamma}(t)$ and the elementary arc length ${\,\text{d}}s$.

$$\int {\boldsymbol v} {\pmb{\cdot}}{\,\text{d}}{\boldsymbol s} := \int\limits^b_a {\boldsymbol v} \bigl({\boldsymbol \gamma}(t)\bigr)^\top \boldsymbol \cdot {\boldsymbol {\dot{\gamma}} }(t) \ {\,\text{d}}t$$

with the vector field ${\boldsymbol v}({\boldsymbol x})$, the curve ${\boldsymbol \gamma}$, and ${\boldsymbol x}, {\boldsymbol v}, {\boldsymbol \gamma} \in {\mathop{\mathbb R}}^n$.

## Surface Integrals

Given a surface ${\boldsymbol \phi}: B \subseteq {\mathop{\mathbb R}}^2 \rightarrow {\mathop{\mathbb R}}^3, (u,w) \mapsto {\boldsymbol \phi}(u,w)$.

$$\iint_{{\boldsymbol \phi}} f {\,\text{d}}O := \iint_B f\bigl({\boldsymbol \phi}(u,w)\bigr) \cdot {\left\lVert{ {\boldsymbol \phi}_u \times {\boldsymbol \phi}_w }\right\rVert} {\,\text{d}}u {\,\text{d}}w$$

with the scalar field $f:D\subseteq {\mathop{\mathbb R}}^3 \rightarrow {\mathop{\mathbb R}}, {\boldsymbol x} \mapsto f({\boldsymbol x})$, the surface ${\boldsymbol \phi}$.

$$\iint_{{\boldsymbol \phi}} {\boldsymbol v} {\pmb{\cdot}}{\,\text{d}}{\boldsymbol O} := \iint_B {\boldsymbol v}\Bigl({\boldsymbol \phi}(u,w)\Bigr)^\top {\pmb{\cdot}}\Bigl( {\boldsymbol \phi}_u \times {\boldsymbol \phi}_w \Bigr) {\,\text{d}}u {\,\text{d}}w$$

with the vector field ${\boldsymbol v}:D\subseteq {\mathop{\mathbb R}}^3 \rightarrow {\mathop{\mathbb R}}^3, {\boldsymbol x} \mapsto {\boldsymbol v}({\boldsymbol x})$, the curve ${\boldsymbol \gamma}$, and ${\boldsymbol x}, {\boldsymbol v}, {\boldsymbol \gamma} \in {\mathop{\mathbb R}}^n$.

## Integral Rules

### Integral Gauß

$$\iiint\limits_V {\text{div}}\; {\boldsymbol v} {\,\text{d}}V = \iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\bigcirc \ {\boldsymbol v} {\pmb{\cdot}}{\,\text{d}}A$$

### Integral Stokes

$$\iint\limits_{A} {\text{rot}}\; {\boldsymbol v} {\,\text{d}}A = \oint\limits_{\partial A} {\boldsymbol v} {\,\text{d}}{\boldsymbol s}$$

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