calculus - Show equality of two integrals: $int_0^inftyfrac{cos x}{1+x}dx = int_0^inftyfrac{sin x}{(1+x)^2}dx$

I need to prove that:

$$\int_0^\infty \frac{\cos x}{1+x} \ dx = \int_0^\infty \frac{\sin x}{(1+x)^2}\ dx$$

but, one converges absolutely whereas the other is not.

I've tried few things like subtitution, integration by parts. I've also tried to use linearty and substract one from the other, hoping the result would be zero.

Yet, none of those worked for me. I guess it's probably demanding some creative subtitution I can't see but I'm not sure.

Answer

You may integrate by parts, obtaining
$$
\int_0^M \frac{\cos x}{1+x} \ dx =\frac{\sin M}{1+M} + \int_0^M \frac{\sin x}{(1+x)^2}\ dx,
$$ then let $M \to \infty$ to get the announced result.