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 \(\vec x, \vec v, \vec \gamma \in \R^n\).