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Question Number 122828 by bemath last updated on 20/Nov/20

Answered by bobhans last updated on 20/Nov/20

$$solve\underset{\pi /4}{\stackrel{\pi /3}{\int }}\frac{\sqrt{\tan x}}{\sin 2x}dx.$$$$Solution:$$$$B\left(x\right)=\int \frac{\sqrt{\tan x}}{2\sin x\cos x}dx$$$$=\frac{1}{2}\int \frac{dx}{\sqrt{\sin x}\cos x\sqrt{\cos x}}$$$$=\frac{1}{2}\int \frac{\sqrt{\cot x}}{\cos {}^{2}x}dx=\frac{1}{2}\int \frac{\sec {}^{2}x}{\sqrt{\tan x}}dx$$$$=\frac{1}{2}\int \frac{d\left(\tan x\right)}{\sqrt{\tan x}}=\sqrt{\tan x}+c$$$$thus\underset{\pi /4}{\stackrel{\pi /3}{\int }}\frac{\sqrt{\tan x}}{\sin 2x}dx=(\sqrt{\tan x}+c){\mid }_{\frac{\pi }{4}}^{\frac{\pi }{3}}$$$$=\sqrt{\sqrt{3}}-1=\sqrt[4]{3}-1.$$

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