Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 159528 by HongKing last updated on 18/Nov/21

Find: 𝛀 =∫_( 0) ^( (𝛑/6)) ((sin(x)βˆ™sin(x + (Ο€/3))βˆ™sin(x + ((2Ο€)/3)))/(sin(3x) + cos(3x))) dx Answer: (Ο€/(48))

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{6}}} {\int}}\frac{\mathrm{sin}\left(\mathrm{x}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\pi}{\mathrm{3}}\right)\centerdot\mathrm{sin}\left(\mathrm{x}\:+\:\frac{\mathrm{2}\pi}{\mathrm{3}}\right)}{\mathrm{sin}\left(\mathrm{3x}\right)\:+\:\mathrm{cos}\left(\mathrm{3x}\right)}\:\mathrm{dx} \\ $$$$\mathrm{Answer}:\:\:\frac{\pi}{\mathrm{48}} \\ $$

Answered by mnjuly1970 last updated on 18/Nov/21

Identity:: sin(x).sin((Ο€/3)βˆ’x).sin((Ο€/3) +x )=(1/4)sin(3x) note: sin(((2Ο€)/3) +x)=sin(Ο€βˆ’((2Ο€)/3)βˆ’x)=sin((Ο€/3) βˆ’x) Ξ© =(1/4)∫_0 ^( (Ο€/6)) ((sin(3x))/(sin(3x)+cos(3x)))dx = (1/(12)){∫ _0^(Ο€/2) (( sin(x))/(sin(x)+cos(x))) dx=_(∫_a ^( b) f(a+bβˆ’x)dx) ^(∫_a ^( b) f(x)dx) βˆ₯ (Ο€/4)} = (1/(12)) ((Ο€/4))=(Ο€/(48)) ...βœ“

$$\:\:\:\mathrm{I}{dentity}::\:\:{sin}\left({x}\right).{sin}\left(\frac{\pi}{\mathrm{3}}βˆ’{x}\right).{sin}\left(\frac{\pi}{\mathrm{3}}\:+{x}\:\right)=\frac{\mathrm{1}}{\mathrm{4}}{sin}\left(\mathrm{3}{x}\right) \\ $$$$\:\:\:{note}:\:\:{sin}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\:+{x}\right)={sin}\left(\piβˆ’\frac{\mathrm{2}\pi}{\mathrm{3}}βˆ’{x}\right)={sin}\left(\frac{\pi}{\mathrm{3}}\:βˆ’{x}\right) \\ $$$$\:\:\:\Omega\:=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \frac{{sin}\left(\mathrm{3}{x}\right)}{{sin}\left(\mathrm{3}{x}\right)+{cos}\left(\mathrm{3}{x}\right)}{dx} \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{12}}\left\{\int\:_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\:{sin}\left({x}\right)}{{sin}\left({x}\right)+{cos}\left({x}\right)}\:{dx}\underset{\int_{{a}} ^{\:{b}} {f}\left({a}+{b}βˆ’{x}\right){dx}} {\overset{\int_{{a}} ^{\:{b}} {f}\left({x}\right){dx}} {=}}\shortparallel\:\:\:\:\:\frac{\pi}{\mathrm{4}}\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{12}}\:\left(\frac{\pi}{\mathrm{4}}\right)=\frac{\pi}{\mathrm{48}}\:\:\:...\checkmark \\ $$

Commented by HongKing last updated on 18/Nov/21

cool thank you so much my dear Ser

$$\mathrm{cool}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Ser} \\ $$

Commented by mnjuly1970 last updated on 18/Nov/21

thanks alot sir..

$$\:\:\:\:{thanks}\:{alot}\:{sir}.. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com