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

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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.