# Integral#

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 $$\diff x$$ was taken to represent an infinitesimally "small piece" of the independent variable $$x$$.

Indefinite integral:

$F(x)=\int f(x)\diff x$

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

Definite integral:

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

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

Example:

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

## Common Integrals#

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

## Trigonometry#

$$\int x \cos(x) \diff x = \cos(x) + x \sin(x)$$
$$\int x \sin(x) \diff x = \sin(x) - x \cos(x)$$
$$\int \sin^2(x) \diff x = \frac12 \bigl(x - \sin(x)\cos(x) \bigr)$$
$$\int \cos^2(x) \diff x = \frac12 \bigl(x + \sin(x)\cos(x) \bigr)$$
$$\int \cos(x)\sin(x) = -\frac12 \cos^2(x)$$

## Line Integrals#

Given a curve $$\vec \gamma:[a,b] \rightarrow \R^n, t \mapsto \vec \gamma(t)$$.

### Scalar field#

$\int\limits_\gamma f \diff s := \int\limits^b_a f\bigl(\vec{\gamma(t)}\bigr) \cdot \norm{\vec{ {\dot{\gamma}} }(t)} \diff t$

with the scalar field $$f(\vec x)$$ along a curve $$\vec \gamma(t)$$ and the elementary arc length $$\diff s$$.

### Vector field#

$\int \vec v \bdot \diff \vec s := \int\limits^b_a \vec v \bigl(\vec \gamma(t)\bigr)^\top \boldsymbol \cdot \vec{ {\dot{\gamma}} }(t) \ \diff t$

with the vector field $$\vec v(\vec x)$$, the curve $$\vec \gamma$$, and $$\vec x, \vec v, \vec \gamma \in \R^n$$.

## Surface Integrals#

Given a surface $$\vec \phi: B \subseteq \R^2 \rightarrow \R^3, (u,w) \mapsto \vec \phi(u,w)$$.

### Scalar field#

$\iint_{\vec \phi} f \diff O := \iint_B f\bigl(\vec \phi(u,w)\bigr) \cdot \norm{ \vec \phi_u \times \vec \phi_w } \diff u \diff w$

with the scalar field $$f:D\subseteq \R^3 \rightarrow \R, \vec x \mapsto f(\vec x)$$, the surface $$\vec \phi$$.

### Vector field#

$\iint_{\vec \phi} \vec v \bdot \diff \vec O := \iint_B \vec v\Bigl(\vec \phi(u,w)\Bigr)^\top \bdot \Bigl( \vec \phi_u \times \vec \phi_w \Bigr) \diff u \diff w$

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

## Integral Rules#

### Integral Gauß#

$\iiint\limits_V \div\; \vec v \diff V = \iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\bigcirc \ \vec v \bdot \diff A$

### Integral Stokes#

$\iint\limits_{A} \rot\; \vec v \diff A = \oint\limits_{\partial A} \vec v \diff \vec s$