Question

What is the integral of `int (cosx)/(x) dx`?

Answer 1

This is one of those integrals that can't be done in terms of elementary functions.
You can do it in terms of infinite series; and you can use various numerical methods to do the definite integral.

The Taylor series expansion of `cos(x)` is

`cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...`

Dividing this by x gives us an infinite series expansion for `cos(x)/x`:

`cos(x)/x = 1/x - x/2! + (x^3)/4! - (x^5)/6! + ...`

And finally, integrating this series term by term gives us a power series expansion for the integral of cos(x)/x:

`int cos(x)/x dx = Ln(x) - (x^2)/(2*2!) + (x^4)/(4*4!) - (x^6)/(6*6!) + ... + c`

where `c` is the integration constant.