...nice calculus... ∫_(0^( ) ) ^( 1) (dx/((x−2)(x^2 (1−x)^3 )^(1/5) )) =?


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Question Number 132741 by mnjuly1970 last updated on 16/Feb/21

$. . . n i c e c a l c u l u s . . .$ $\int_{0^{}}^{1} \frac{d x}{(x - 2) {(x^{2} {(1 - x)}^{3})}^{\frac{1}{5}}} = ?$
Commented by MJS_new last updated on 16/Feb/21

$I get - 2^{1 / 10} π \sqrt{1 - \frac{\sqrt{5}}{5}} \approx - 2.50340750$ $using the substitution t = {(\frac{2 (1 - x)}{x})}^{1 / 5}$
Commented by MJS_new last updated on 16/Feb/21

$yes of course . forgot the ‘ ‘ -^{″}$
Commented by Dwaipayan Shikari last updated on 16/Feb/21

$Y e s s i r! B u t i t h i n k t h e r e s u l t s h o u l d b e - v e$
Commented by mnjuly1970 last updated on 16/Feb/21

$b r a v o m r M j$ $t h a n k y o u s o m u c h . . .$
	Answered by Dwaipayan Shikari last updated on 16/Feb/21

	$- \int_{0}^{1} t^{- \frac{3}{5}} {(1 - t)}^{- \frac{2}{5}} {(1 + t)}^{- 1} d t 1 - x = t$ $_{2} F_{1} (a, b; c; z) = \frac{Γ (c)}{Γ (c - b) Γ (b)} \int_{0}^{1} x^{b - 1} {(1 - x)}^{c - b - 1} {(1 - z x)}^{- a} d x$ $b - 1 = - \frac{3}{5} \Rightarrow b = \frac{2}{5}$ $c - b - 1 = \frac{- 2}{5} \Rightarrow c = 1$ $a = 1 z = - 1$ $- Γ (\frac{3}{5}) Γ (\frac{2}{5})_{2} F_{1} (1, \frac{2}{5}, 1, - 1) = - \int_{0}^{1} t^{- \frac{3}{5}} {(1 - t)}^{- \frac{2}{5}} {(1 + t)}^{- 1} d t$ $= - \frac{2 \sqrt{2} π}{\sqrt{5 - \sqrt{5}}}_{2} F_{1} (1, \frac{2}{5}, 1; - 1)$
	Commented by mnjuly1970 last updated on 16/Feb/21

	$m e r c e y$ $g r a t e f u l . . .$ $h y p e r g e o m e t r e y f u n c t i o n . . .$
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