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substitution notation
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The following are two commonly used substitution notations for calculating definite integrals with the antiderivative:
- $\int_a^b f(x)\,dx = \left[F(x)\right]_a^b$
- $\int_a^b f(x)\,dx = F(x)|_a^b$
Here, the right hand sides mean the difference $F(b)-F(a)$ . For example, one has $$\int_1^2\frac{1}{x}\,dx \;=\; \left[\ln x\right]_1^2.$$ In Finland (only?) the corresponding notation is $$\int_1^2\frac{1}{x}\,dx \;=\; \sijoitus{1}{\quad 2}\ln x$$ which may be somewhat better; it is read in same manner as the definite integral notation, ``sijoitus 1:stä 2:een ln x'' (literally: ``substitution from 1 to 2 ln x''). The position of the substitution symbol in front of the function to be substituted is perhaps more natural in
the sense that the symbol has an operator character (as e.g. the summing symbol). One of benefits of the Finnish notation is that one can comfortably clarify in it which is the variable to be substituted (as in the sum notation), e.g. in the case $$\int_0^\pi\sin{tx}\,dt \;=\; -\frac{1}{x}\sijoitus{t = 0}{\quad \pi}\cos{tx}.$$
The notation $$\sijoitus{a}{\quad b}\!F(x) \;:=\; F(b)-F(a)$$ is extended also to such cases as $$\sijoitus{a}{\quad\infty}\!F(x) \;:=\; \lim_{b\to\infty}\sijoitus{a}{\quad b}\!F(x).$$
Formulae
- $\sijoitus{a}{b}\!F(x) \;=\; -\!\sijoitus{b}{a}\!F(x)$
- $\sijoitus{a}{b}\!kF(x) \;=\; k\!\sijoitus{a}{b}\!F(x)$
- $\sijoitus{a}{b}\![F_1(x)+\ldots+F_n(x)] \;=\; \sijoitus{a}{b}\!F_1(x)+\ldots+\sijoitus{a}{b}\!F_n(x)$
- $\int_a^b u(x)\,v'(x)\,dx \;=\; \sijoitus{a}{b}\!u(x)\,v(x) -\int_a^b u'(x)\,v(x)\,dx$
Note. There are in Finland also some other ``national'', unofficial mathematical notations used in universities, e.g. $$-\!\!\!\ni\!\!\!-$$ which means `such that'. For example, one may write $$\forall\, x \in \mathbb{Z}\; \exists\, y \in \mathbb{Z}\;\; -\!\!\!\ni\!\!\!- \;\; x\!+\!y = 0.$$
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"substitution notation" is owned by pahio.
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See Also: Hermite polynomials, area under Gaussian curve, sine integral at infinity, tractrix, Fourier sine and cosine series, proof of closed differential forms on a simple connected domain, Taylor series of arcus tangent, perimeter of astroid, potential of hollow ball, volume of ellipsoid, Archimedes' cylinders in cube, centre of mass of half-disc, evolute, Laplace transform of sine integral, Laplace transform of  , volume of spherical cap and spherical sector, calculating the solid angle of disc, value of Riemann zeta function at  , Laplace transform of derivative, uncertainty principle, generalisation of Gaussian integral, using convolution to find Laplace transform, relative of exponential integral, relative of cosine integral, integral related to arc sine, example of improper integral, orthogonality of Legendre polynomials, application of logarithm series, arc length example, errors can cancel each other out, convergence of integrals, area bounded by arc and two lines, orthogonality of Laguerre polynomials
This object's parent.
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Cross-references: variable, summing, operator, function, substitution, difference, right, antiderivative, definite integrals
This is version 15 of substitution notation, born on 2005-03-15, modified 2009-05-22.
Object id is 6880, canonical name is SubstitutionNotation.
Accessed 2892 times total.
Classification:
| AMS MSC: | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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