A Nice Integral (4)

Calculus Level 3

Evaluate ∫ 0 π / 2 sin ⁡ x sin ⁡ x + cos ⁡ x d x \large \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} \, dx

π 4 \frac{\pi}{4} π \pi π 2 \frac{\pi}{2} 2 π 2\pi

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Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Chew-Seong Cheong
Aug 11, 2016

I = ∫ 0 π 2 sin ⁡ x sin ⁡ x + cos ⁡ x d x Using the identity ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = 1 2 ( ∫ 0 π 2 sin ⁡ x sin ⁡ x + cos ⁡ x d x + ∫ 0 π 2 sin ⁡ ( π 2 − x ) sin ⁡ ( π 2 − x ) + cos ⁡ ( π 2 − x ) d x ) = 1 2 ( ∫ 0 π 2 sin ⁡ x sin ⁡ x + cos ⁡ x d x + ∫ 0 π 2 cos ⁡ x cos ⁡ x + sin ⁡ x d x ) = 1 2 ∫ 0 π 2 sin ⁡ x + cos ⁡ x sin ⁡ x + cos ⁡ x d x = 1 2 ∫ 0 π 2 d x = x 2 ∣ 0 π 2 = π 4 \begin{aligned} I & = \int_0^\frac \pi 2 \frac {\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx \quad \quad \small \color{#3D99F6}{\text{Using the identity }\int_a^b f(x) \ dx = \int_a^b f(a+b - x) \ dx} \\ & = \frac 12 \left(\int_0^\frac \pi 2 \frac {\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx + \int_0^\frac \pi 2 \frac {\sqrt{\sin \left(\frac \pi 2 - x\right)}}{\sqrt{\sin \left(\frac \pi 2 - x\right)}+\sqrt{\cos \left(\frac \pi 2 - x\right)}} dx \right) \\ & = \frac 12 \left(\int_0^\frac \pi 2 \frac {\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx + \int_0^\frac \pi 2 \frac {\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} dx \right) \\ & = \frac 12 \int_0^\frac \pi 2 \frac {\sqrt{\sin x}+\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx \\ & = \frac 12 \int_0^\frac \pi 2 dx \\ & = \frac x2 \bigg|_0^\frac \pi 2 = \boxed{\dfrac \pi 4}\end{aligned}

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Nice solution as usual, got my vote.

Hana Wehbi - 4 years, 10 months ago

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Thanks. I have edited your problem. Because sin ⁡ \sin is a function, it should have a "\" in front in LaTex just like "sqrt" for \sqrt{} . But you need to leave a space before you enter x x . Functions should not be in italic like tan ⁡ , cos ⁡ , ln ⁡ , gcd ⁡ \tan, \cos, \ln, \gcd all with "\" in front. Italic is for variable and constant such as x , y , z , a , b , c x, y, z, a, b, c .

Chew-Seong Cheong - 4 years, 10 months ago

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Thank you.

Hana Wehbi - 4 years, 10 months ago

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