# Show that ∫ a 0 √ X 3 a 3 − X 3 D X = a √ X γ ( 5 6 ) γ ( 1 3 ) - Applied Mathematics 2

Sum

Show that int_0^asqrt(x^3/(a^3-x^3))dx=a(sqrtxgamma(5/6))/(gamma(1/3))

#### Solution

Let "I"=int_0^asqrt(x^3/(a^3-x^3))dx

Put x^3=a^3t  =>x=at^(1/3)
Diff. w.r.t. x ,

dx=a/3t^((-2)/3) dt

Limits becomes →[𝟎,𝟏]

"I"=int_0^1(t)^(3/2)(1-t)^(3/2)t^((-2)/3)a/3dt

=a/3int_0^1t^(5/6)(1-t)^(3/2)dt

=a/3beta(5/6,3/2)

"I"=a(sqrtxgamma(5/6))/(gamma(1/3) …………{ from the definition of beta function}

Concept: Differentiation Under Integral Sign with Constant Limits of Integration
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