...... calculus.....(III)...... evaluate:: 𝛗=^(???) ∫_0 ^( (π/2)) ∫_0 ^( x) ((cos(y))/( (√(((π/2)−x)((π/2)−y)))))dydx


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Question Number 138114 by mnjuly1970 last updated on 10/Apr/21

$. . . . . . c a l c u l u s . . . . . (I I I) . . . . . .$ $e v a l u a t e ::$ $ϕ \overset{? ? ?}{=} \int_{0}^{\frac{π}{2}} \int_{0}^{x} \frac{c o s (y)}{\sqrt{(\frac{π}{2} - x) (\frac{π}{2} - y)}} d y d x$
Commented by Kamel last updated on 10/Apr/21

Commented by mnjuly1970 last updated on 10/Apr/21

$t h a n k s a l o t m r . . k a m e l$
	Answered by mnjuly1970 last updated on 10/Apr/21

	$ϕ = \int_{0}^{\frac{π}{2}} \int_{y}^{\frac{π}{2}} \frac{c o s (y)}{\sqrt{(\frac{π}{2} - x) (\frac{π}{2} - y)}} d x d y$ $= 2 \int_{0}^{\frac{π}{2}} \frac{c o s (y)}{\sqrt{\frac{π}{2} - y}} {[- \sqrt{\frac{π}{2} - x}]}_{y}^{\frac{π}{2}} d y$ $= 2 \int_{0}^{\frac{π}{2}} c o s (y) d y = 2 s i n (\frac{π}{2}) = 2 . . . ✓$
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