Some Physically Useful Definite Integrals

This is just a sampling of some of the definite integrals which appear in physical problems


Application:
molecular velocity distribution

Other definite integrals associated with Gaussian distribution.
Squares of trig functions
Indefinite integrals
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Definite Integrals Associated with Gaussian Distributions

In physical systems which can be modeled by a Gaussian distribution, one sometimes needs to obtain the average or expectation value for physical quantities. If these properties depend on x, then they can be integrated to find the average value. For the first five powers of x, the integrals have the following forms:


Applications:
Average molecular kinetic energy
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Definite Integrals of the Squares of Trig Functions

In dealing with sinusoidal functions associated with waves and those associated with AC electric circuits, the squares of trig functions appear. If they are averaged, then the average of the square of a trig function is found to be 1/2, so long as you are taking an integer number of quarter periods.


Applications:
AC electric power
Radiated power from surface
Propagated power in EM wave
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