Question Number 128316 by mnjuly1970 last updated on 08/Jan/21
$$\:\:\:\:\:\:\:\:\:\:\:\:{nice}\:\:{calculus} \\ $$$$\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({x}\right)}{{x}^{\mathrm{2}} }{dx}=? \\ $$$$ \\ $$
Answered by mindispower last updated on 26/Jan/21
$${by}\:{part} \\ $$$$\Omega=\mathrm{3}\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}\right){sin}^{\mathrm{2}} \left({x}\right)}{{x}}{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left(\mathrm{2}{x}\right){sin}\left({x}\right)}{{x}}{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}\right)−{cos}\left(\mathrm{3}{x}\right)}{{x}}{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}{Re}\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{ix}} −{e}^{\mathrm{3}{ix}} }{{x}}{dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{ix}} −{e}^{\mathrm{3}{ix}} }{{x}}{e}^{−{xs}} {dx}={f}\left({s}\right),{s}\geqslant\mathrm{0} \\ $$$$\left.{f}'\left({s}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{{x}\left({i}−{s}\right)} −{e}^{{x}\left(\mathrm{3}{i}−{s}\right.} \right){dx} \\ $$$$=\left[\frac{{e}^{{x}\left({i}−{s}\right)} }{{i}−{s}}−\frac{{e}^{{x}\left(\mathrm{3}{i}−{s}\right)} }{\mathrm{3}{i}−{s}}\right]_{\mathrm{0}} ^{\infty} =\frac{\mathrm{1}}{{s}−{i}}+\frac{\mathrm{1}}{\mathrm{3}{i}−{s}}={f}'\left({s}\right) \\ $$$${f}\left({s}\right)={ln}\left(\frac{{s}−{i}}{{s}−\mathrm{3}{i}}\right)+{c} \\ $$$$\underset{{s}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{ix}} −{e}^{\mathrm{3}{ix}} }{{x}}{e}^{−{xs}} {dx}\rightarrow\mathrm{0} \\ $$$$\Rightarrow{c}=\mathrm{0} \\ $$$${f}\left(\mathrm{0}\right)={ln}\left(\frac{\mathrm{1}}{\mathrm{3}}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}\right)−{cos}\left(\mathrm{3}{x}\right)}{{x}}{dx} \\ $$$$\Rightarrow\Omega=−\frac{\mathrm{3}{ln}\left(\mathrm{3}\right)}{\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{4}}{ln}\left(\frac{\mathrm{1}}{\mathrm{27}}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$