How to Integrate (Cos[x])/x Using Taylor Series Expansion

[plato2000]

Homework Statement

∫$\frac{Cos(x)}{x}$ dx

Homework Equations

Taylor series expansion for Cos(x)

The Attempt at a Solution

I have used Taylor series to find the product of (1/x) * (cos[x]). After integration i get

In[x] - x^2/8 + x^4/96 + x^6/4320+...

I don't know what to do next, is that the answer, or there is a way of finding the function represented by the series above.

 

[Ray Vickson]

Homework Statement

∫$\frac{Cos(x)}{x}$ dx

Homework Equations

Taylor series expansion for Cos(x)

The Attempt at a Solution

I have used Taylor series to find the product of (1/x) * (cos[x]). After integration i get

In[x] - x^2/8 + x^4/96 + x^6/4320+...

I don't know what to do next, is that the answer, or there is a way of finding the function represented by the series above.

Up to an additive constant the integral is a non-elementary function called Ci(x). There is no finite, closed-form expression for Ci(x) that involves only elementary functions such as powers, roots, exponentials, trig functions, etc. That is provable: it is not just that nobody has been smart enough to find the formula, but, rather, that it has been rigoroursly proven that no such formula can possibly exist!

 

[plato2000]

Thank you very much.

 

[vanhees71]

Well, you are allowed to integrate the series term by term (think about why!). That shows that the integral exists. What doesn't exist is an expression in terms of elementary funktions (i.e., polynomials and exponential functions and their inverses).