Antiderative
- Thread starter shahar
- Start date
topsquark
Senior Member
- Joined
- Aug 27, 2012
- Messages
- 2,307
There is no closed form for this indefinite integral. You can, perhaps, get some exact answers for definite integrals (I haven't looked to see if there are any simple ones for this.)
If you need a form to use, expand sin(x) into a Taylor series about the point of interest, then integrate the series. You won't be able to write the summation as a function, but it will give you a form you can use to estimate it.
-Dan
You have the sine function divided by a different function, and the Order of Operations.
You should write that as sin(x)/x at a minimum of characters.
Otis
Elite Member
- Joined
- Apr 22, 2015
- Messages
- 4,592
Hi shahar. Dan explained why (there's no formula), and the reason the calculator returned an "error" response is because it hasn't been programmed with algorithms to handle the situation.
Several computer algebra systems define a named function to handle the form [imath]\int{\frac{sin(x)}{x}}[/imath]. For example, my copy of MapleV would show the first derivative of f(x)=2*x*sin(x^2-1)/(x^2-1) as Si(x^2-1), and integrating Si(x^2-1) would return f(x). One could do calculations with Complex numbers or look at graphs with Real values.
I seem to remember using Si(x) in college, maybe, with Fourier transforms. (For the most part, those synapses have blissfully dissolved away.)
[imath]\;[/imath]
Otis
Elite Member
- Joined
- Apr 22, 2015
- Messages
- 4,592
Whoops, I typed that inversely.
I meant, integrating f(x) gives Si(x) and differentiating Si(x) returns f(x).I checked wolframalpha.com, and they also use the name Si(x). They even have a named function for the ratio of sin(x) to x: sinc(x). We can graph my example with plot {Si(x^2-1), 2*x*sinc(x^2-1)}.
