What is the use of Dirichlet Integral? How can I find the value of 
$$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$
using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied the basics of contour integration...
Further, how can I generalise it to the following? $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^ndx$$

HINT:

We provide herein only an outline of the approach to evaluating the integral of interest using contour integration.

Step 1:  First note that $\frac{\sin z}{z}$ is an even function.  Therefore, 
$$\int_0^\infty\frac{\sin^5 x}{x^5}\,dx=\frac12 \int_{-\infty}^\infty\frac{\sin^5 x}{x^5}\,dx$$
Step 2:  Use Euler's Identity along with the Binomial Theorem to write 
$$\sin^5 x=\frac{e^{i5z}}{32i}\,\sum_{k=0}^5\binom{5}{k}(-1)^k\,e^{-i2kz}$$
Step 3:  Note for $n>0$ ($n<0$) we have from the residue theorem
$$\begin{align}
\oint_C \frac{e^{inz}}{z^5}\,dz&=\text{sgn}(n)\,2\pi i \text{Res}\left(\frac{e^{inz}}{z^5},z=0\right)\\\\
&=2\pi i \text{sgn}(n) \left(\frac{n^4}{4!}\right)
\end{align}$$
where $C$ is a contour consisting of (i) the real-line segments from $(-R,0)$ to $(-\epsilon,0)$ and from $(\epsilon,0)$ to $(R,0)$, (ii) a semi-circle in the lower-half (upper-half) plane, centered at the origin with radius $\epsilon$, and (iii) a semi-circle in the upper-half (lower-half) plane, centered at the origin with radius $R$.
Then, note finally that 
$$\lim_{R\to \infty}\lim_{\epsilon \to 0}\oint_C\frac{e^{inz}}{z^5}\,dz=\text{PV}\left(\int_{-\infty}^\infty\frac{e^{inx}}{x^5}\,dx\right)+\text{sgn}(n)\pi i \frac{n^4}{4!}$$
The rest should be straightforward.
