Question-197637

Question Number 197637 by SLVR last updated on 25/Sep/23

Commented by SLVR last updated on 25/Sep/23

$${kindly}\:{help}\:{me}\:{sir} \\ $$

Commented by SLVR last updated on 25/Sep/23

sir..i mean ∫_0 ^π (x/(1−sinxcosx))dx=? it was asked to prove as ((5π^2 )/(6(√3)))...kindly help me

$${sir}..{i}\:{mean}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{x}}{\mathrm{1}−{sinxcosx}}{dx}=? \\ $$$${it}\:{was}\:{asked}\:{to}\:{prove}\:{as} \\ $$$$\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{6}\sqrt{\mathrm{3}}}…{kindly}\:{help}\:{me} \\ $$

Commented by SLVR last updated on 26/Sep/23

$${kindly}\:\:\:{help}\:{me} \\ $$

Answered by witcher3 last updated on 27/Sep/23

I=∫_0 ^π (x/(1−sin(x)cos(x)))dx =∫_0 ^π ((π−x)/(1+sin(x)cos(x))) 2I=∫_0 ^π (π/(1+sin(x)cos(x)))dx+∫_0 ^π x((sin(2x))/(1−sin^2 (x)cos^2 (x)))dx ∫_0 ^π (π/(1+sin(x)cos(x)))dx =π∫_0 ^(π/2) (dx/(1+sin(x)cos(x)))+(dx/(1−sin(x)cos(x))) π∫_0 ^(π/2) (1+tan^2 (x))((dx/(1+tan^2 (x)+tan(x)))+(dx/(1−tan(x)+tan^2 (x)))) =π∫_0 ^∞ ((1/((y^2 +y+1)))+(1/(y^2 −y+1)))dy =π[(2/( (√3)))tan^(−1) ((y+(1/2)).(2/( (√3))))+(2/( (√3)))tan^(−1) ((2/( (√3)))(y−(1/2)))]_0 ^∞ π((2/( (√3)))((π/2)+(π/2)))=(2/( (√3)))π^2 ∫_0 ^π ((xsin(2x))/(1−sin^2 (x)cos^2 (x))) ∫((sin(2x))/(1−sin^2 (x)cos^2 (x))) =4∫((sin(2x))/(4−sin^2 (2x))) =2∫((sin(y))/(4−(1−cos^2 (y))))=2∫((sin(y))/(3+cos^2 (y))) =−2∫((d(cos(y)))/(3+cos^2 (y))) =−(2/( (√3)))tan^(−1) (((cos(y))/( (√3))))+c ∫_0 ^π ((xsin(2x))/(3+cos^2 (2x))=[_0 ^π x.((−2)/( (√3)))tan^(−1) (((cos(2x))/( (√3))))] +(2/( (√3)))∫_0 ^π tan^(−1) (((cos(2x))/( (√3))))dx =−(π^2 /(3(√3)))+(2/( (√3)))∫_0 ^(π/2) tan^(−1) (((cos(2x))/( (√3))))+tan^(−1) (((cos(2((π/2)+x)))/( (√3))))dx =−(π^2 /(3(√3)))+(2/( (√3)))∫_0 ^(π/2) tan^(−1) (((cos(2x))/( (√3))))+tan^(−1) (−((cos(2x))/( (√3)))))dx=−(π^2 /(3(√3))) 2I=((2π^2 )/( (√3)))−(π^2 /(3(√3)))⇔I=((5π^2 )/(6(√3)))

$$\mathrm{I}=\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{x}}{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)}\mathrm{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi} \frac{\pi−\mathrm{x}}{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)} \\ $$$$\mathrm{2I}=\int_{\mathrm{0}} ^{\pi} \frac{\pi}{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)}\mathrm{dx}+\int_{\mathrm{0}} ^{\pi} \mathrm{x}\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)}\mathrm{dx} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\pi}{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)}\mathrm{dx} \\ $$$$=\pi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{dx}}{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)}+\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)} \\ $$$$\pi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)\right)\left(\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)+\mathrm{tan}\left(\mathrm{x}\right)}+\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{tan}\left(\mathrm{x}\right)+\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)}\right) \\ $$$$=\pi\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\left(\mathrm{y}^{\mathrm{2}} +\mathrm{y}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} −\mathrm{y}+\mathrm{1}}\right)\mathrm{dy} \\ $$$$=\pi\left[\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\left(\mathrm{y}+\frac{\mathrm{1}}{\mathrm{2}}\right).\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\right)+\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left(\mathrm{y}−\frac{\mathrm{1}}{\mathrm{2}}\right)\right)\right]_{\mathrm{0}} ^{\infty} \\ $$$$\pi\left(\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left(\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{2}}\right)\right)=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\pi^{\mathrm{2}} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{xsin}\left(\mathrm{2x}\right)}{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)} \\ $$$$\int\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{cos}^{\mathrm{2}} \left(\mathrm{x}\right)} \\ $$$$=\mathrm{4}\int\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\mathrm{4}−\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2x}\right)} \\ $$$$=\mathrm{2}\int\frac{\mathrm{sin}\left(\mathrm{y}\right)}{\mathrm{4}−\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \left(\mathrm{y}\right)\right)}=\mathrm{2}\int\frac{\mathrm{sin}\left(\mathrm{y}\right)}{\mathrm{3}+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{y}\right)} \\ $$$$=−\mathrm{2}\int\frac{\mathrm{d}\left(\mathrm{cos}\left(\mathrm{y}\right)\right)}{\mathrm{3}+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{y}\right)} \\ $$$$=−\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{y}\right)}{\:\sqrt{\mathrm{3}}}\right)+\mathrm{c} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{xsin}\left(\mathrm{2x}\right)}{\mathrm{3}+\mathrm{cos}^{\mathrm{2}} \left(\mathrm{2x}\right.}=\left[_{\mathrm{0}} ^{\pi} \mathrm{x}.\frac{−\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{3}}}\right)\right] \\ $$$$+\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\int_{\mathrm{0}} ^{\pi} \mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{3}}}\right)\mathrm{dx} \\ $$$$=−\frac{\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}}+\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{3}}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{2}\left(\frac{\pi}{\mathrm{2}}+\mathrm{x}\right)\right)}{\:\sqrt{\mathrm{3}}}\right)\mathrm{dx} \\ $$$$\left.=−\frac{\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}}+\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{3}}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(−\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\:\sqrt{\mathrm{3}}}\right)\right)\mathrm{dx}=−\frac{\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}} \\ $$$$\mathrm{2I}=\frac{\mathrm{2}\pi^{\mathrm{2}} }{\:\sqrt{\mathrm{3}}}−\frac{\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}}\Leftrightarrow\mathrm{I}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{6}\sqrt{\mathrm{3}}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Commented by witcher3 last updated on 05/Oct/23