Question Number 200862 by sonukgindia last updated on 25/Nov/23
Answered by Mathspace last updated on 25/Nov/23
$$\Phi \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{5}{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{10}} }{dx}=_{{x}={t}^{\frac{\mathrm{1}}{\mathrm{10}}} } \frac{\mathrm{1}}{\mathrm{10}}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{5}\:{t}^{\frac{\mathrm{1}}{\mathrm{5}}} }{\mathrm{1}+{t}}{t}^{\frac{\mathrm{1}}{\mathrm{10}}−\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{\frac{\mathrm{3}}{\mathrm{10}}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\pi}{{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{10}}\right)} \\ $$$$=\frac{\pi}{\mathrm{2}{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{10}}\right)} \\ $$