Answer to Question #281789 in Calculus for chaitu

Question

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

Answer 1

10.

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

where beta function:

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

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

9.

Dirichlet’s Integral:

$\int^{\infin}_0\frac{sinx}{x}dx=\pi/2$

for gamma function:

$\Gamma(1-z)\Gamma(z)=\pi/sin\pi z$