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

Question

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.

Answer 1

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

Answer 2

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

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

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