# Solving Integrals using summations

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1. Nov 5, 2014

1. The problem statement, all variables and given/known data
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!

2. Relevant equations

$\sin(x)$

3. The attempt at a solution

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.

Last edited by a moderator: Nov 5, 2014
2. Nov 5, 2014

### HallsofIvy

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

3. Nov 5, 2014

### 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 $\int \sum f_n = \sum \int f_n.$

Last edited: Nov 5, 2014
4. Nov 5, 2014