0-pi-2-sin-6-cos-4-d-

Question Number 139353 by mohammad17 last updated on 26/Apr/21

\int_{0}^{\frac{π}{2}} {s i n}^{6} θ {c o s}^{4} θ d θ

Answered by Dwaipayan Shikari last updated on 26/Apr/21

\int_{0}^{\frac{π}{2}} {s i n}^{2 α - 1} θ {c o s}^{2 β - 1} θ d θ = \frac{Γ (α) Γ (β)}{2 Γ (α + β)}

\int_{0}^{\frac{π}{2}} {s i n}^{6} θ {c o s}^{4} θ d θ = \frac{Γ (\frac{7}{2}) Γ (\frac{5}{2})}{2 Γ (6)} = \frac{\frac{5}{2} . \frac{3}{2} . \frac{1}{2} . \frac{3}{2} . \frac{1}{2} Γ^{2} (\frac{1}{2})}{240}

= \frac{3}{2^{9}} π = \frac{3 π}{512}

Answered by ajfour last updated on 26/Apr/21

I = \int_{0}^{π / 2} \sin^{6} θ \cos^{4} θ d θ

= \int_{0}^{π / 2} \cos^{6} θ \sin^{4} θ d θ

2 I = \int_{0}^{π / 2} \sin^{4} θ \cos^{4} θ d θ

32 I = \int_{0}^{π / 2} \sin^{4} 2 θ d θ

128 I = \int_{0}^{π / 2} {(1 - \cos 4 θ)}^{2} d θ

256 I = \int_{0}^{π / 2} (2 - 4 \cos 4 θ + 1 + \cos 8 θ) d θ

256 I = \frac{3 π}{2}

I = \frac{3 π}{512}

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