Integrating sin(x)/x using Integration by Parts

[jackalsniper]

i'm having troubles to integrate \int\frac{sin x}{x} dx
could anyone help me with hints? i tried using integration by parts, but i see it as a never ending chain, sin will change to cos... and x^-1 will become -x^-2 and so on

 

[gabbagabbahey]

That integral can't be expressed in closed form in terms of elementary functions. SinIntegral

 

[jackalsniper]

i don't really understand that article, can u explain it please?

 

[gabbagabbahey]

i don't really understand that article, can u explain it please?

I doubt that I can provide a much better explanation than what is given in the article. It should be pretty self-explanatory provided you have at least a 2nd year university level math understanding.

Are you sure you are trying to evaluate the correct integral? Is this part of a larger problem? Have you come across special functions before?

 

[jackalsniper]

no, it's a question that says,
compute the integral of (sinx / x) dx by means of
a. the trapezoidal rule
b. a three term gausian quadrature formulae.
c. simpson's 1/3 rule
d. determine the integral analytically.

i was thinking that part d. uses integration by parts to solve

 

[gabbagabbahey]

Is it a definite integral? If so, what are the integration limits?

 

[jackalsniper]

integrate from 0 to 1, sorry i forgot to include that

 

[gabbagabbahey]

integrate from 0 to 1, sorry i forgot to include that

In that case, use the power series expansion for sine and integrate it term by term. You can either leave your answer as an infinite sum, or explicitly sum up the first 3-4 non-zero terms. Summing to order x^7 gives a result accurate to 6 decimal places.

It's too bad it wasn't from zero to infinity, that produces a nice exact answer

 

[jackalsniper]

u mean like \int\stackrel{1}{0} \frac{sin x}{x} dx = x - \frac {x\stackrel{3}{}} {(3 \cdot 3!)} + \frac {x\stackrel{5}{}} {(5 \cdot 5!)} - \frac {x\stackrel{7}{}} {(7 \cdot 7!)} + - ...?

 

[gabbagabbahey]

u mean like \int\stackrel{1}{0} \frac{sin x}{x} dx = x - \frac {x\stackrel{3}{}} {(3 \cdot 3!)} + \frac {x\stackrel{5}{}} {(5 \cdot 5!)} - \frac {x\stackrel{7}{}} {(7 \cdot 7!)} + - ...?

Don't forget to substitute in the limits!

\int_0^1 \frac{sin x}{x} dx = \left[ x - \frac {x\stackrel{3}{}} {(3 \cdot 3!)} + \frac {x\stackrel{5}{}} {(5 \cdot 5!)} - \frac {x\stackrel{7}{}} {(7 \cdot 7!)} +...\right]_0^1

You can write it as an infinite sum; \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)(2n+1)!} or just evaluate the first few terms.

 

[jackalsniper]

ok, thanks for the help, i appreciate it very much