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Question-137707




Question Number 137707 by mohammad17 last updated on 05/Apr/21
Answered by Dwaipayan Shikari last updated on 05/Apr/21
∫_0 ^(π/2) (√(tanθ)) dθ  =∫_0 ^(π/2) sin^(1/2) θ  cos^(−(1/2)) θ dθ  =((Γ((3/4))Γ((1/4)))/2)=(((√2)π)/2)=(π/( (√2)))
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{tan}\theta}\:{d}\theta \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\frac{\mathrm{1}}{\mathrm{2}}} \theta\:\:{cos}^{−\frac{\mathrm{1}}{\mathrm{2}}} \theta\:{d}\theta \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{2}}=\frac{\sqrt{\mathrm{2}}\pi}{\mathrm{2}}=\frac{\pi}{\:\sqrt{\mathrm{2}}} \\ $$

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