Solving Integrals using summations

[Amad27]

Homework Statement

Many places I have seen when solving integrals you change a lot of it into sums.

http://math.stackexchange.com/quest...dfrac-tan-x1m2-tan2x-mathrmdx/1006076#1006076

Is just an example.

So in general, how do you solve integrals (CLOSED FORM) by using series?

Thanks!

(for example integrate $\displaystyle \int_{0}^{4\pi} \sin(x) dx$ using SERIES)?

Thanks!

Homework Equations

$\sin(x)$

The Attempt at a Solution

[/B]
I am a beginner, with no expertise in this area, so I asked the question. I don't know any methods to start with. Any help will be appreciated.

 

[HallsofIvy]

Do you know what the MacLaurin series (Taylor series centered on x= 0) is for sin(x)?

 

[FeDeX_LaTeX]

Tonelli's theorem says that if $f_{n}(x) \geq 0 \text{ } \forall n, x$, then we can interchange the sum and the integral as follows:

$$\sum \int f_{n}(x) dx = \int \sum f_{n}(x) dx.$$

For general $f_n,$ Fubini's theorem says that if either $\int \sum |f_n|$ or $\sum \int |f_n|$ converge, then $\int \sum f_n = \sum \int f_n.$

 

[Mark44]

Amad27,
When you write a LaTeX expression, please use the correct tags. For itex, the closing tag is /itex, not \itex (in brackets). I fixed your post here and the LaTeX in your other post about limits.

You can check that what you have done is correctly formatted by clicking the PREVIEW button.