Hello,
I need help in solving the following real integral using the residue theorem.
$\displaystyle \int_0^\infty \frac{sin^2 x}{x^2}dx$.
What starting complex function should I use? and what contour?
Thank you
One way:
(a) Consider $\displaystyle \int_C\dfrac{e^{iz}dz}{z},\quad C=\gamma_1\cup\gamma_2\cup \gamma_3\cup\gamma_4$
$\displaystyle \begin{Bmatrix}\gamma_1(t)=t,\;t\in[ \epsilon,R]\\\gamma_2(t)=Re^{it},\;t\in [0,\pi]\\\gamma_3(t)=t,\;t\in [-R,- \epsilon]\\ \gamma_4(t)= \epsilon e^{(\pi-t)i},\;t\in [0,\pi]\end{matrix}$
Using the residue theorem you'll obtain $\displaystyle \int_0^{+\infty}\dfrac{\sin x \;dx}{x}=\dfrac{\pi}{2}$
(b) Using the integration by parts method, you'll obtain
$\displaystyle \int_0^{+\infty}\dfrac{\sin^2 x \;dx}{x^2}=\int_0^{+\infty}\dfrac{\sin 2x \;dx}{x}$
(c) Using the substitution $\displaystyle t=2x$ you'll obtain
$\displaystyle \int_0^{+\infty}\dfrac{\sin 2x \;dx}{x}=\int_0^{+\infty}\dfrac{\sin t \;dt}{t}=\dfrac{\pi}{2}$