I try to solve this indefinite integral:
\begin{equation}
\int\,\mathrm{sinc}^{2}\left(ax\right)\,\cos\left(bx\right)\,\,\mathrm{d}x
\end{equation}
whith $\mathrm{sinc}\left(x\right) = \frac{\sin\left(x\right)}{x}$. I tried to do integratin by parts using the D-I-method / tabular method. However, this resulted in very lengthy and complicated expressions, independent which part is considered for differentiation and for integration. Do you know some other strategy how to obtain a solution? Of course I checked WolframAlpha and seeing that there exists a solution lets me feel at least certain that an analytic solution exists.
https://www.wolframalpha.com/input?i=integrate+sinc%28a*x%29%5E%282%29+*+cos%28b*x%29+dx
Your help is appreciated, thank you 🙂
Best Answer
How about a reduction formula?
(I took the case $a=b=1$ of the original question.)
Consider integrals of the form $$ J(a,b,c) = \int\frac{\sin^a(x)\cos^b(x)}{x^c}\;dx , $$ where $c$ is a positive integer. We want a "reduction formula" to evalute this in terms of similar integrals with smaller value of $c$.
Integrate by parts to get something like $$ J(a,b,c) = -\frac{\sin^a(c)\cos^b(x)}{(c-1)x^{c-1}} +\frac{1}{c-1}\int\left(\frac{(a+b)\sin^{a-1}(x)\cos^{b+1}(x)}{x^{c-1}} -\frac{b\sin^{a-1}(x)\cos^{b-1}(x)}{x^{c-1}}\right)\;dx \\ =-\frac{\sin^a(c)\cos^b(x)}{(c-1)x^{c-1}} +\frac{a+b}{c-1}J(a-1,b+1,c-1)-\frac{b}{c-1}J(a-1,b-1,c-1) . $$ Starting values $J(a,b,1)$ involve sine integral $\operatorname{Si}$ and/or cosine integral $\operatorname{Ci}$.