Is Substituting sinx/lnx Legal?

[romsofia]

Is this "legal"?

$${sinx= \sum^\inf_0 \frac{-1^nx^{2n+1}}{(2n+1)!}}$$

Now, let's say we take the integral of this: $${\int sinx = \int \sum^\inf_0 \frac{-1^nx^{2n+1}}{(2n+1)!} = \sum^\inf_0 \frac{-1^n}{(2n+1)!} \int x^{2n+1}}$$

Which we will get: $${\sum^\inf_0 \frac{-1^n}{(2n+1)!} \frac{x^{2n+2}}{2n+2}+C}$$

Which of course is the power series for cosx (as we expected).

The reason why I'm asking this is, am I allowed to make this substitution for sinx/lnx?

I.e: $${\int \frac{sinx}{lnx} = \int \sum^\inf_0 \frac{-1^nx^{2n+1}}{(2n+1)!}*\frac{1}{lnx} = \sum^\inf_0 \frac{-1^n}{(2n+1)!} \int \frac{x^{2n+1}}{lnx} = \sum^\inf_0 \frac{-1^n}{(2n+1)!} Ei((2n+2)lnx)+C}$$

Thanks for your time and help.

EDIT: I think mute might've come to this conclusion a while back (I'd have to check my old threads).

 

[jedishrfu]

this is legal math as the integration of x is independent of the summation symbol n.