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Question-168338




Question Number 168338 by infinityaction last updated on 08/Apr/22
Answered by Mathspace last updated on 08/Apr/22
∫_0 ^∞   ((sin^2 (ax))/x^2 )dx    a>0  =_(ax=t)   ∫_0 ^∞   ((sin^2 (t))/(((t/a))^2 ))(dt/a)  =a∫_0 ^∞   ((sin^2 t)/t^2 )dt  by parts  ∫_0 ^∞   ((sin^2 t)/t^2 )dt=[−(1/t)sin^2 t]_0 ^(→∞)   +∫_0 ^∞  (1/t)(2sint.cost)dt  =0+∫_0 ^∞   ((sin(2t))/t)dt  (2t=z)  =2∫_0 ^∞  ((sinz)/z)(dz/2)=∫_0 ^∞  ((sinz)/z)dz=(π/2) ⇒  ∫_0 ^∞  ((sin^2 (ax))/x^2 )dx=((πa)/2)  if a<0 we get the same result   due?to  x→((sin^2 (ax))/x^2 ) is even
$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\mathrm{2}} }{dx}\:\:\:\:{a}>\mathrm{0} \\ $$$$=_{{ax}={t}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} \left({t}\right)}{\left(\frac{{t}}{{a}}\right)^{\mathrm{2}} }\frac{{dt}}{{a}} \\ $$$$={a}\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} {t}}{{t}^{\mathrm{2}} }{dt}\:\:{by}\:{parts} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{2}} {t}}{{t}^{\mathrm{2}} }{dt}=\left[−\frac{\mathrm{1}}{{t}}{sin}^{\mathrm{2}} {t}\right]_{\mathrm{0}} ^{\rightarrow\infty} \\ $$$$+\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{t}}\left(\mathrm{2}{sint}.{cost}\right){dt} \\ $$$$=\mathrm{0}+\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left(\mathrm{2}{t}\right)}{{t}}{dt}\:\:\left(\mathrm{2}{t}={z}\right) \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:\frac{{sinz}}{{z}}\frac{{dz}}{\mathrm{2}}=\int_{\mathrm{0}} ^{\infty} \:\frac{{sinz}}{{z}}{dz}=\frac{\pi}{\mathrm{2}}\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi{a}}{\mathrm{2}} \\ $$$${if}\:{a}<\mathrm{0}\:{we}\:{get}\:{the}\:{same}\:{result}\: \\ $$$${due}?{to}\:\:{x}\rightarrow\frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\mathrm{2}} }\:{is}\:{even} \\ $$$$ \\ $$$$ \\ $$
Commented by infinityaction last updated on 08/Apr/22
thank you sir
$${thank}\:{you}\:{sir} \\ $$

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