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Question-122828




Question Number 122828 by bemath last updated on 20/Nov/20
Answered by bobhans last updated on 20/Nov/20
 solve ∫_(π/4) ^(π/3)  ((√(tan x))/(sin 2x)) dx .    Solution :   B(x)= ∫ ((√(tan x))/(2sin x cos x)) dx    = (1/2)∫ (dx/( (√(sin x)) cos x (√(cos x))))   = (1/2)∫ ((√(cot x))/(cos^2 x)) dx = (1/2)∫ ((sec^2 x)/( (√(tan x)))) dx   = (1/2)∫ ((d(tan x))/( (√(tan x)))) = (√(tan x)) + c   thus ∫_(π/4) ^(π/3)  ((√(tan x))/(sin 2x)) dx = ( (√(tan x)) + c )∣_(π/4) ^(π/3)    = (√((√3) )) − 1 = (3)^(1/4)  − 1.
$$\:{solve}\:\underset{\pi/\mathrm{4}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\:.\: \\ $$$$\:{Solution}\::\: \\ $$$${B}\left({x}\right)=\:\int\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{2sin}\:{x}\:\mathrm{cos}\:{x}}\:{dx}\: \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:{x}}\:\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:{x}}} \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\sqrt{\mathrm{cot}\:{x}}}{\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{\mathrm{sec}\:^{\mathrm{2}} {x}}{\:\sqrt{\mathrm{tan}\:{x}}}\:{dx} \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{{d}\left(\mathrm{tan}\:{x}\right)}{\:\sqrt{\mathrm{tan}\:{x}}}\:=\:\sqrt{\mathrm{tan}\:{x}}\:+\:{c}\: \\ $$$${thus}\:\underset{\pi/\mathrm{4}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\:=\:\left(\:\sqrt{\mathrm{tan}\:{x}}\:+\:{c}\:\right)\mid_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \\ $$$$\:=\:\sqrt{\sqrt{\mathrm{3}}\:}\:−\:\mathrm{1}\:=\:\sqrt[{\mathrm{4}}]{\mathrm{3}}\:−\:\mathrm{1}.\:\: \\ $$

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