0
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Ahh, why didn't I see this! Thanks for the pointer. $\endgroup$ – genap Feb 1 '16 at 20:30
2
$\begingroup$

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

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

$$\frac44\int_0^\infty\frac{\sin(u)}{u}\,du=\frac{\pi}{2}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.