How would one go about computing

$$\int_0^\infty \frac{\sin(4x)}{x}$$

without any background in complex analysis (e.g. using strictly calculus)?

I know that

$$\int_0^\infty \frac{\sin(x)}{x} = \frac{\pi}{2}$$

and I was thinking I might be able to use this identity somehow.

The integral can be transformed using double angle identities into

$$\int_0^\infty \frac{4\sin(x)\cos(x)\cos(2x)}{x}$$

but I'm not sure how to go forward from there.


Your integral is identical to the one you want to use, as a substitution $u=4x$ shows.

  • $\begingroup$ Ahh, why didn't I see this! Thanks for the pointer. $\endgroup$ – genap Feb 1 '16 at 20:30


Substitute $u=4x$ and $du=4dx$:



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