Find the value of the folloing integral. determinant ((( 𝛀=∫_0 ^( (𝛑/2)) (( 1)/(1 + (( cosx))^(1/3) )) dx = ? )))


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Question Number 208176 by mnjuly1970 last updated on 07/Jun/24

$F i n d t h e v a l u e o f t h e$ $f o l l o i n g i n t e g r a l .$ $\begin{matrix} Ω = \int_{0}^{\frac{π}{2}} \frac{1}{1 + \sqrt[3]{c o s x}} d x = ? \end{matrix}$
Commented by Frix last updated on 07/Jun/24

$See question 208140$
	Answered by Berbere last updated on 07/Jun/24
	$(1/(1+((cos(x)))^(1/3) ))=Σ(−1)^n cos^(n/3) (x) Ω=Σ_(n≥0) ∫_0 ^(π/2) (−1)^n cos^(2((n/6)+(1/2))−1) (x)sin^(2((1/2))−1) (x)dx =Σ_(n≥0) (((−1)^n )/2)β((n/6)+(1/2),(1/2))=Σ_(n≥0) (((−1)^n )/2)((Γ((n/6)+(1/2))Γ((1/2)))/(Γ((n/6)+1))) N=∪_(k∈{0,1,2,3,4,5)) (6N+k) =Σ_(k=0) ^5 Σ_(m=0) ^∞ (((−1)^(6m+k) )/2).((Γ(m+(k/6)+(1/2)))/(Γ(m+(k/6)+1))) Γ((1/2)) =Γ((1/2))Σ_(k=0) ^5 (((−1)^k )/2)Σ_(m≥0) (((Γ(m+(k/6)+(1/2)))/(Γ(m)))/((Γ(m+(k/6)+1))/(Γ(m)))).Γ(m+1).(1^m /(m!)) =((√π)/2)Σ_(k=0) ^5 _2 F_1 ((k/6)+(1/2),1;(k/6)+1;[1])$
	$\frac{1}{1 + \sqrt[3]{c o s (x)}} = Σ {(- 1)}^{n} {c o s}^{\frac{n}{3}} (x)$ $Ω = \sum_{n ⩾ 0} \int_{0}^{\frac{π}{2}} {(- 1)}^{n} {c o s}^{2 (\frac{n}{6} + \frac{1}{2}) - 1} (x) {s i n}^{2 (\frac{1}{2}) - 1} (x) d x$ $= \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{2} β (\frac{n}{6} + \frac{1}{2}, \frac{1}{2}) = \sum_{n ⩾ 0} \frac{{(- 1)}^{n}}{2} \frac{Γ (\frac{n}{6} + \frac{1}{2}) Γ (\frac{1}{2})}{Γ (\frac{n}{6} + 1)}$ $N = \underset{k \in {0, 1, 2, 3, 4, 5)}{\cup} (6 N + k)$ $= \underset{k = 0}{\sum^{5}} \underset{m = 0}{\sum^{\infty}} \frac{{(- 1)}^{6 m + k}}{2} . \frac{Γ (m + \frac{k}{6} + \frac{1}{2})}{Γ (m + \frac{k}{6} + 1)} Γ (\frac{1}{2})$ $= Γ (\frac{1}{2}) \underset{k = 0}{\sum^{5}} \frac{{(- 1)}^{k}}{2} \sum_{m ⩾ 0} \frac{\frac{Γ (m + \frac{k}{6} + \frac{1}{2})}{Γ (m)}}{\frac{Γ (m + \frac{k}{6} + 1)}{Γ (m)}} . Γ (m + 1) . \frac{1^{m}}{m!}$ $= \frac{\sqrt{π}}{2} \underset{k = 0}{\sum^{5}}_{2} F_{1} (\frac{k}{6} + \frac{1}{2}, 1; \frac{k}{6} + 1; [1])$
	Commented by Frix last updated on 07/Jun/24

	$Nice!$
	Commented by Berbere last updated on 10/Jun/24

	$t h a n k Y o u s i r H a v e a n i c e D a y$
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