Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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.

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com