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Integration using parametric derivatives
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Integration using parametric derivatives
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{{Short description|Method which uses known Integrals to integrate derived functions}}{{multipleissues|{{refimprove|date=July 2019}}{{context|date=June 2019}}}}In calculus, integration by parametric derivatives, also called parametric integration,JOURNAL, Zatja, Aurel J., Parametric Integration Techniques {{!, Mathematical Association of America |website=www.maa.org |date=December 1989 |volume=Mathematics Magazine |url=https://www.maa.org/sites/default/files/268948443847.pdf |access-date=23 July 2019}} is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.- the content below is remote from Wikipedia
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Statement of the theorem
By using The Leibniz integral rule with the upper and lower bounds fixed we get thatfrac{d}{dt}left(int_a^b f(x,t)dxright)=int_a^b frac{partial}{partial t} f(x,t)dxIt is also true for non-finite bounds.Examples
Example One: Exponential Integral
For example, suppose we want to find the integral
int_0^infty x^2 e^{-3x} , dx.
Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it. However, we may also evaluate this by starting with a simpler integral and an added parameter, which in this case is t = 3:
int_0^infty e^{-tx} , dx = frac{1}{t},
we can differentiate both sides twice with respect to t (not x) in order to add the factor of x2 in the original integral.
int_0^infty x^2 e^{-3x} , dx = frac{2}{3^3} = frac{2}{27}.
Example Two: Gaussian Integral
Starting with the integral int^infty_{-infty} e^{-x^2t}dx=frac{sqrtpi}{sqrt t},taking the derivative with respect to t on both sides yieldsbegin{align}&frac{d}{dt}int^infty_{-infty} e^{-x^2t}dx=frac{d}{dt}frac{sqrtpi}{sqrt t}&-int^infty_{-infty} x^2 e^{-x^2t} = -frac{sqrt pi}{2}t^{-frac{3}{2}}&int^infty_{-infty} x^2e^{-x^2t}= frac{sqrt{pi}}{2}t^{-frac{3}{2}}end{align}.In general, taking the n-th derivative with respect to t gives usint^infty_{-infty} x^{2n}e^{-x^2t}= frac{(2n-1)!!sqrt pi}{2^n}t^{-frac{2n+1}{2}}.Example Three: A Polynomial
Using the classical int x^t dx=frac{x^{t+1}}{t+1} and taking the derivative with respect to t we getint ln(x)x^t= frac{ln(x)x^{t+1}}{t+1} - frac{x^{t+1}}{(t+1)^2}.Example Four: Sums
The method can also be applied to sums, as exemplified below. Use the Weierstrass factorization of the sinh function:frac{sinh (z)}{z}=prod_{n=1}^infty left(frac{pi^2 n^2 + z^2}{pi^2 n^2}right).Take the logarithm:ln(sinh (z)) - ln(z)=sum_{n=1}^infty lnleft(frac{pi^2 n^2 + z^2}{pi^2 n^2}right).Derive with respect to z:coth(z) - frac{1}{z}= sum^infty_{n=1}frac{2z}{z^2+pi^2n^2}.Let w=frac{z}{pi}:frac{1}{2}frac{coth(pi w)}{pi w} - frac{1}{2}frac{1}{z^2}=sum^infty_{n=1}frac{1}{n^2+w^2}.References
{{reflist}}External links
WikiBooks: Parametric_Integration{{mathanalysis-stub}}- content above as imported from Wikipedia
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