Integrating sinx/x: Solving a Common Homework Question

[elabed haidar]

Homework Statement

simple question what is the integral of sinx/x i really tried but i couldn't find the trick

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

There is a very simple "trick":
$$\int\frac{sin(x)}{x}dx= Si(x)+ C$$
because "Si(x)", the "sine integral
http://mathworld.wolfram.com/SineIntegral.html
is defined to be that integral.

sin(x)/x cannot be integrated in terms of elementary functions.

 

[Ray Vickson]

To expand a bit on the answer given by HallsofIvy: the indefinite integral of f(x) = sin(x)/x cannot be done in finitely many terms involving elementary functions, such as powers, roots, trig functions, exponentials, logarithms, etc. This is a rigorously proven *theorem*. It means that even if you allow yourself 100 billion pages you could not write the answer "explicitly" on that amount of paper, using only elementary functions. Of course, you could immediately write an infinite series answer, but we are not counting that as "explicit". Again, this is a *theorem*; it is not just that nobody has been smart enough to know how to do it, but rather, that it is impossible to do.

RGV

 

[elabed haidar]

thank you very much so that was the trick people are talking about lol thank you all but thank you ray vickson for the explanation