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apply beta function to solve definite integral x^4/1+x^6x=0, x=8

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jigsfix | Student, Undergraduate | (Level 3) eNoter

Posted January 24, 2012 at 5:17 PM via web

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apply beta function to solve definite integral x^4/1+x^6

x=0, x=8

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted January 24, 2012 at 6:27 PM (Answer #1)

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You need to use `beta`  function to solve the definite integral, hence you should come up with the substitution `x^6 = t =gt 6x^5dx = dt =gt dx = (dt)/(6x^5)`

`` `x = root(6) t =gt x = t^(1/6) =gt x^5 = t^(5/6)`

`x = 0 =gt t = 0`

You need to write the integral using the variable t such that:

`(1/6) int_0^oo t^(-5/6)(1+t)^(-1)dt = (1/6)beta(5/6,1/6)`

`beta(5/6,1/6) = (Gamma(5/6)*Gamma(1/6))/(Gamma(5/6+1/6))`

`` `Gamma(5/6) = int_0^oo x^(5/6 - 1)e^(-x) dx`

`Gamma(5/6+1/6) = Gamma(1) = 1`

`` `(1/6)(Gamma(5/6)*Gamma(1/6)) = 2pi/6 = pi/3`

Hence, using `beta ` function, the definite integral `int_0^oo (x^4dx)/(1+x^6) = pi/3` .

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