calculate-0-1-x-1-x-6-dx-

Question Number 168303 by Mathspace last updated on 07/Apr/22

c a l c u l a t e \int_{0}^{1} x \sqrt{1 - x^{6}} d x

Answered by peter frank last updated on 07/Apr/22

check Qn 168231

Answered by Mathspace last updated on 08/Apr/22

t h i s i n t e g r a l i s r e s o l u b l e

w e d o t h e c h a n g e m e n t x = t^{\frac{1}{6}} \Rightarrow

\int_{0}^{1} x \sqrt{1 - x^{6}} d x = \frac{1}{6} \int_{0}^{1} t^{\frac{1}{6}} {(1 - t)}^{\frac{1}{2}} t^{\frac{1}{6} - 1} d t

= \frac{1}{6} \int_{0}^{1} t^{\frac{1}{3} - 1} {(1 - t)}^{\frac{3}{2} - 1} d t

w e k n o w B (p, q) = \int_{0}^{1} t^{p - 1} {(1 - t)}^{q - 1} d t

= \frac{Γ (p) . Γ (q)}{Γ (p + q)} \Rightarrow

I = \frac{1}{6} B (\frac{1}{3}, \frac{3}{2}) = \frac{1}{6} \times \frac{Γ (\frac{1}{3}) . Γ (\frac{3}{2})}{Γ (\frac{1}{3} + \frac{3}{2})}

Γ (\frac{3}{2}) = Γ (\frac{1}{2} + 1) = \frac{1}{2} Γ (\frac{1}{2}) = \frac{\sqrt{π}}{2}

\Rightarrow I = \frac{\sqrt{π}}{12} . \frac{Γ (\frac{1}{3})}{Γ (\frac{11}{6})}