Answer to Question #281789 in Calculus for chaitu

Question #281789


9. Find the expression for ∫︁∫︁∫︁∫︁

V

x

l−1y

m−1z

n−1 dx by DZ(Dirichlet’s Integral) in

the form of Gamma integrals, here V is the region x ≥ 0, y ≥ 0, z ≥ 0 and

x + y + z ≤ a.

Hint/Ans: Γ(l)Γ(m)Γ(n)

Γ(l + m + n + 1)

a

l+m+n


10. Evaluate ∫︁

1



1

1 − x4

dx


1
Expert's answer
2021-12-23T14:16:24-0500

10.

01dx/1x4=B(1/4,1/2)/4=1.311\int^1_0dx/\sqrt{1-x^4}=B(1/4,1/2)/4=1.311


where beta function:

B(x,y)=01tx1(1t)y1dtB(x,y)=\int^1_0t^{x-1}(1-t)^{y-1}dt


B(1/4,1/2)=01t3/4(1t)1/2dtB(1/4,1/2)=\int^1_0t^{-3/4}(1-t)^{-1/2}dt


9.

Dirichlet’s Integral:


0sinxxdx=π/2\int^{\infin}_0\frac{sinx}{x}dx=\pi/2


for gamma function:

Γ(1z)Γ(z)=π/sinπz\Gamma(1-z)\Gamma(z)=\pi/sin\pi z


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