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Definition

An integral is a function FF of which a given function ff 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 dx\,\text{d}x was taken to represent an infinitesimally “small piece” of the independent variable xx.

Indefinite integral:

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

with the function ff of a real variable xx.

Definite integral:

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

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

Example

Given f(x)=x2f(x) = x^2, then 12x2dx=[13x3]12=8313=73=213\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)CF(x) - C f(x)f(x) f(x)f'(x)
1q+1xq+1\frac{1}{q+1}x^{q+1} xqx^q qxq1qx^{q-1}
2ax33\frac{2\sqrt{ax^3}}{3} ax\sqrt{ax} a2ax\frac{a}{2\sqrt{ax}}
xln(ax)xx\ln(ax) -x ln(ax)\ln(ax) $\textstyle \frac{1}{x}$
1a2eax(ax1)\frac{1}{a^2} e^{ax}(ax- 1) xeaxx \cdot e^{ax} eax(ax+1)e^{ax}(ax+1)
axln(a)\frac{a^x}{\ln(a)} axa^x axln(a)a^x \ln(a)
cos(x)-\cos(x) sin(x)\sin(x) cos(x)\cos(x)
sin(x)\sin(x) cos(x)\cos(x) sin(x)-\sin(x)
ln|cos(x)|-\ln \left\vert{\cos(x)}\right\vert tan(x)\tan(x) 1cos2(x)\frac{1}{\cos^2(x)}
ln|sin(x)|\ln \left\vert{\sin(x)}\right\vert cot(x)\cot(x) 1sin2(x)\frac{-1}{\sin^2(x)}
xarcsin(x)+1x2x\arcsin (x)+\sqrt{1-x^2} arcsin(x)\arcsin(x) 11x2\frac{1}{\sqrt{1-x^2}}
xarccos(x)1x2x\arccos (x)-\sqrt{1-x^2} arccos(x)\arccos(x) 11x2-\frac{1}{\sqrt{1-x^2}}
xarctan(x)12ln|1+x2|x\arctan (x)-\frac{1}{2} \ln \left\vert{1+ x^2}\right\vert arctan(x)\arctan(x) 11+x2\frac{1}{1+x^2}
xarctan(x)+12ln|1+x2|x\arctan (x)+\frac{1}{2} \ln \left\vert{1+ x^2}\right\vert arccot(x)\text{arccot}(x) 11+x2-\frac{1}{1+x^2}
cosh(x)\cosh(x) sinh(x)\sinh(x) cosh(x)\cosh (x)
sinh(x)\sinh(x) cosh(x)\cosh(x) sinh(x)\sinh (x)

Trigonometry

xcos(x)dx=cos(x)+xsin(x)\int x \cos(x) \,\text{d}x = \cos(x) + x \sin(x)
xsin(x)dx=sin(x)xcos(x)\int x \sin(x) \,\text{d}x = \sin(x) - x \cos(x)
sin2(x)dx=12(xsin(x)cos(x))\int \sin^2(x) \,\text{d}x = \frac12 \bigl(x - \sin(x)\cos(x) \bigr)
cos2(x)dx=12(x+sin(x)cos(x))\int \cos^2(x) \,\text{d}x = \frac12 \bigl(x + \sin(x)\cos(x) \bigr)
cos(x)sin(x)=12cos2(x)\int \cos(x)\sin(x) = -\frac12 \cos^2(x)

Line Integrals

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

γfds:=abf(𝛄(t))𝛄̇(t)dt\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(𝐱)f(\boldsymbol x) along a curve 𝛄(t)\boldsymbol \gamma(t) and the elementary arc length ds\,\text{d}s.

𝐯d𝐬:=ab𝐯(𝛄(t))𝛄̇(t)dt\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 𝐯(𝐱)\boldsymbol v(\boldsymbol x), the curve 𝛄\boldsymbol \gamma, and $\boldsymbol x, \boldsymbol v, \boldsymbol \gamma \in \R^n$.


Surface Integrals

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

𝛟fdO:=Bf(𝛟(u,w))𝛟u×𝛟wdudw\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 𝛟\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 𝛄\boldsymbol \gamma, and $\boldsymbol x, \boldsymbol v, \boldsymbol \gamma \in \R^n$.


Integral Rules

Integral Gauß

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

Integral Stokes

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