Math Question and Answers


Question and Answers Forum

All Questions Topic List
Integration Questions
Previous in All Question Next in All Question
Previous in Integration Next in Integration
Question Number 120774 by mnjuly1970 last updated on 02/Nov/20

$. . . a d v a n c e d c a l c u l u s . . .$ $p r o v e t h a t ::$ $Ω = \int_{0}^{1} \frac{l n (x)}{\sqrt[3]{1 - x^{3}}} d x \overset{? ? ?}{=} - \frac{π}{3 \sqrt{3}} (l n (3) + \frac{π}{3 \sqrt{3}})$ $. . . m . n . 1970 . . .$
	Answered by mindispower last updated on 02/Nov/20

	$l e t x^{3} = t \Rightarrow d x = t^{- 2 \frac{1}{3}} \frac{d t}{3}$ $\Rightarrow Ω = \frac{1}{3} \int_{0}^{1} \frac{l n (t^{\frac{1}{3}})}{{(1 - t)}^{\frac{1}{3}}} . t^{- 2 \frac{1}{3}} \frac{d t}{3}$ $Ω = \frac{1}{9} \int_{0}^{1} {(1 - t)}^{- \frac{1}{3}} t^{- 2 \frac{1}{3}} l n (t) d t$ $l n (t) = \frac{\partial}{\partial a} x^{a} ∣_{x = 0}$ $Ω = \partial_{a} \frac{1}{9} \int_{0}^{1} {(1 - t)}^{\frac{1}{3}} t^{- \frac{2}{3} + a} d t$ $9 Ω = \frac{\partial}{\partial a} β (\frac{2}{3}, \frac{1}{3} + a) ∣_{a = 0}$ $\frac{\partial}{\partial y} β (x, y) = β (x, y) [Ψ (y) - Ψ (x + y)]$ $\frac{\partial β}{\partial a} = β (\frac{2}{3}, \frac{1}{3} + a) [Ψ (\frac{1}{3} + a) - Ψ (1 + a)]$ $9 Ω = β (\frac{1}{3}, \frac{2}{3}) [Ψ (\frac{1}{3}) - Ψ (1)]$ $G a u s s e t h e o r e m$ $Ψ (\frac{p}{q}) = - γ - l n (2 q) - \frac{π}{2} c o t (\frac{p π}{q}) + 2 \underset{n = 1}{\sum^{[\frac{q - 1}{2}]}} c o s (\frac{2 π n p}{q}) l n (s i n (\frac{n π}{q}))$ $Ψ (\frac{1}{3}) = - γ - l n (6) - \frac{π}{2} . \frac{1}{\sqrt{3}} - l n (\frac{\sqrt{3}}{2})$ $= - γ - l n (3 \sqrt{3}) - \frac{π}{2 \sqrt{3}}$ $Ψ (1) = - γ$ $β (\frac{1}{3}, \frac{2}{3}) = \frac{Γ (\frac{1}{3}) Γ (\frac{2}{3})}{Γ (2)} = . Γ (\frac{1}{3}) Γ (\frac{2}{3}) = \frac{π}{s i n (\frac{π}{3})}$ $= \frac{2 π}{\sqrt{3}}$ $9 Ω = \frac{2 π}{\sqrt{3}} (- γ - l n (3 \sqrt{3}) - \frac{π}{2 \sqrt{3}} + γ)$ $= \frac{2 π}{\sqrt{3}} (- l n (3 \sqrt{3}) - \frac{π}{2 \sqrt{3}})$ $Ω = - \frac{π}{27} (- \sqrt{3} l n (27) - π) = - \frac{π}{27} (π + \sqrt{3} l n (27))$
	Commented by mnjuly1970 last updated on 03/Nov/20

	$t h a n k s a l o t s i r m i n d s p o w e r$ $p e a c e b e u p o n y o u . . .$
	Commented by mindispower last updated on 03/Nov/20

	$w i t h e p l e a s u r s i r h a v e a n i c e D a y$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum