Question Number 138860 by qaz last updated on 19/Apr/21
$${Prove}::\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} −\mathrm{sin}\:^{\mathrm{3}} {x}}{{x}^{\mathrm{5}} }{dx}=\frac{\mathrm{13}}{\mathrm{32}}\pi \\ $$
Answered by Dwaipayan Shikari last updated on 19/Apr/21
$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} −{sin}^{\mathrm{3}} {x}}{{x}^{\mathrm{5}} }{dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} −\left(\mathrm{3}{sinx}−{sin}\mathrm{3}{x}\right)/\mathrm{4}}{{x}^{\mathrm{5}} }{dx} \\ $$$${J}\left({a}\right)+{G}\left(\mathrm{3}\right)=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{4}{x}^{\mathrm{3}} −\mathrm{3}{sin}\left({ax}\right)}{{x}^{\mathrm{5}} }+\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\mathrm{3}{x}}{{x}^{\mathrm{5}} }{dx} \\ $$$${J}''\left({a}\right)=\frac{\mathrm{3}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({ax}\right)}{{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{3}{a}^{\mathrm{2}} }{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({t}\right)}{\:{t}^{\mathrm{3}} }{dt}=\frac{\mathrm{3}{a}^{\mathrm{2}} }{\mathrm{4}}.\frac{\pi}{\mathrm{2}\Gamma\left(\mathrm{3}\right){sin}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\right)} \\ $$$$=−\frac{\mathrm{3}{a}^{\mathrm{2}} \pi}{\mathrm{16}}\:\Rightarrow{J}\left({a}\right)=−\frac{{a}^{\mathrm{4}} \pi}{\mathrm{64}}+{Ka}+{C}\:\: \\ $$$${K}=\mathrm{0}\:\:\:\:\:{C}=\mathrm{0} \\ $$$${G}\left(\mathrm{3}\right)=\frac{\mathrm{3}^{\mathrm{4}} }{\mathrm{4}}.\frac{\pi}{\mathrm{2}\Gamma\left(\mathrm{5}\right){sin}\left(\frac{\mathrm{5}\pi}{\mathrm{2}}\right)}=\frac{\mathrm{81}\pi}{\mathrm{192}}=\frac{\mathrm{27}\pi}{\mathrm{64}} \\ $$$${J}''\left(\mathrm{1}\right)+{G}\left(\mathrm{3}\right)=\frac{−\pi}{\mathrm{64}}+\frac{\mathrm{27}\pi}{\mathrm{64}}=\frac{\mathrm{13}\pi}{\mathrm{32}} \\ $$
Commented by qaz last updated on 19/Apr/21
$${thank}\:{you}\:{sir} \\ $$
Commented by mathmax by abdo last updated on 20/Apr/21
$$\mathrm{what}\:\mathrm{mean}\:\mathrm{J}\left(\mathrm{a}\right)\:\mathrm{sir}… \\ $$