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

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

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 $\boldsymbol x, \boldsymbol v, \boldsymbol \gamma \in \R^n$.

Surface Integrals

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

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

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

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

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