Question

Solve ${\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{x^{3}cosx}{si{n}^{2}x}dx$

Answer 1

We have

${\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{x^3\cos x}{{\sin }^{2}x}dx$

$I={\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}{x}^3\mathrm{cscxcotxdx}$

$I=x^3{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\mathrm{cscxcotxdx}-{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\left(\frac{d{x}^3}{dx}\mathrm{cscxcotxdx}\right)dx$

$I=x^3{{(-\mathrm{cscx})}_{-\frac{\pi }{3}}}^{\frac{\pi }{3}}-{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}4x^3(-\csc x)dx$

$I=x^3{{(-\mathrm{cscx})}_{-\frac{\pi }{3}}}^{\frac{\pi }{3}}+4\left[x^3{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\left(\csc x\right)dx-{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\left(\frac{d{x}^3}{dx}{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\left(\csc x\right)dx\right)dx\right]+C$

$I=-x^3(\csc \frac{\pi }{3}+\csc \frac{\pi }{3})+x^3{{(-\ln (\csc x+\cot x))}_{-\frac{\pi }{3}}}^{\frac{\pi }{3}}-12{\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}{x}^2{{(-\ln (\csc x+\cot x))}_{-\frac{\pi }{3}}}^{\frac{\pi }{3}}dx$

$I=-----------------$

$soon.$

Hence, this is the answer.