Question

integral

Answer 1

no actually got lost on it

Answer 2

\[\int\limits \sin(x^2) dx\]

Answer 3

can you give me a hint?

Answer 4

Fresnel integral: http://en.wikipedia.org/wiki/Fresnel_integral

I'm afraid there's no closed form :/

Answer 5

sure, but that form that includes the sum notation, how would I come to it?
by doing integration by parts, many times?

Answer 6

"No closed form" generally means you won't be able to find an antiderivative, at least not in terms of elementary functions (i.e. polynomials, trig functions, exponentials, logarithms etc)

Answer 7

yes, I see... like for example

e^x/x would be something like sigma, infinity terms, i=0

for (n! e^x / x(x)^n)

Answer 8

I mean n=0, not i=0

Answer 9

and I want to know how to come to this not-closed form

Answer 10

and that e^x/x I would get the sigma, but integrating by parts many times,
differentiating the bottom x each time

Answer 11

But in this case, (I mean,) how would I get the sigma notation answer?

Answer 12

I can elaborate if you want, I sound pretty confusing....

Answer 13

use power series of sin(x) to express sin(x^2) as an infinite sum

Answer 14

http://i.stack.imgur.com/oiv7g.png

Answer 15

I see the power series for sin(x), so sin(x^2) would be
x^3/ 3 - x^7 /3!7 + x^11/5!11 + x^15/7!.....
right?

Answer 16

maybe use the sum notation..
\[\begin{align}\sin (\color{Red}{x}) &=\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\color{Red}{x}^{2n+1} \\~\\
\sin (\color{Red}{x^2}) &=\cdots \end{align}\]

Answer 17

just replace x by x^2

Answer 18

\[=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{4x+2}\]

Answer 19

I was never good at those, but this should fine, after you chewed it for me:D

Answer 20

:) these are fun
Careful.. x is your function variable and "n" is the temporary index variable in sum

Answer 21

I am aware, tnx

Answer 22

maybe use the sum notation..
\[\begin{align}\sin (\color{Red}{x}) &=\sum\limits_{\color{blue}{n}=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\color{Red}{x}^{2\color{blue}{n}+1} \\~\\
\sin (\color{Red}{x^2}) &=\sum\limits_{\color{blue}{n}=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\color{Red}{x}^{4\color{blue}{n}+2} \end{align}\]

Answer 23

I will try to get a tutor, to go over this entire power series thing. I haven't spent time studying it in school.... nice colors, btw.

are you able to put this inline?

Answer 24

or it's impossible?

Answer 25

yeah, the integration part, as much as I could get, I got.

Answer 26

Nice integral for the day !

Answer 27

pain in my a... jK

Answer 28

right click on the reply and click "Show Math As" ---> "Tex commands"

it shows you the latex code..

Answer 29

I thought that integration by parts would remove any questions..

Answer 30

Angle -Angle -Side

Answer 31

tnx.

Answer 32

If I see anyone put an inline sigma, then I would make sure to copy the command and put in in my latex

Answer 33

Btw, do you know anyone who knows how to do it, so that I can msg?

Answer 34

I mean to put inline sigmas, and integrals but so that they look nice, not messed up.

Answer 35

Power series are powerful because the integral of infinite sum equals the sum of integral of each term with the same radius of convergence

Answer 36

All that means is that you can switch \(\large \int\) and \(\large \sum\) signs when you're dealing with power series (conditions apply)

Answer 37

not this way, \(\ \sum_{n=c_1}^{c_2}f(n) \)

Answer 38

but a normal, nice looking on?

Answer 39

copy paste below exact code

```
\[\begin{align}

\sin (\color{Red}{x})
&=\sum\limits_{\color{blue}{n}=0}^{\infty}
\frac{(-1)^n}{(2n+1)!}\color{Red}{x}^{2\color{blue}{n}+1} \\~\\

\sin (\color{Red}{x^2})
&=\sum\limits_{\color{blue}{n}=0}^{\infty}
\frac{(-1)^n}{(2n+1)!}\color{Red}{x}^{4\color{blue}{n}+2} \\~\\

\end{align}\]
```

Answer 40

`\(\sum_{n=1}^\infty\)` gives \(\sum_{n=1}^\infty\), whereas `\(\displaystyle\sum_{n=1}^\infty\)` gives \(\displaystyle\sum_{n=1}^\infty\).

Answer 41

oh cool, tnx!

Answer 42

for an integral you would also add \displaystyle?

Answer 43

and for a limit too?

Answer 44

Yes, though you also have the option of the `\limits` command if you want to keep the small size.

Example: `\(\sum\limits_{n=1}^\infty\)` yields \(\sum\limits_{n=1}^\infty\).

Answer 45

\(\large \displaystyle \lim_{x \rightarrow a} f(x) \)

Answer 46

\(\large \displaystyle \lim_{x \rightarrow a} f(x) \) works!

Answer 47

oh, it's not hard...

Answer 48

How did you know to do this, this is a very nice trick, (the displaystyle) !

Answer 49

you wont be needing them if you use `\[ \]` instead of the inline code `\( \)`
only the inline latex doesn't know where to put the limits

Answer 50

limits work with inline latex as well

Answer 51

`\[ \sum_a^b \]` produces below
\[ \sum_a^b \]

Answer 52

\(\large \displaystyle \lim_{x \rightarrow a} f(x) \) oh

\(\large \displaystyle \int\limits_{a}^{b} f(x)~dx \) snap

Answer 53

yes you mayh use `\limits` explicitly in inline code..

Answer 54

The limits `\limits` stuff also works with other commands, like `\bigcup`, `\bigcap`, `\sup`, etc:
\(\bigcup_{x\in X}\iff\bigcup\limits_{x\in X}\)

\(\bigcap_{x\in X}\iff\bigcap\limits_{x\in X}\)

\(\sup_{x\in X}f(x)\iff\sup\limits_{x\in X}f(x)\)

Answer 55

sup? don't even want to learn that... omg...

Answer 56

hey, sup?

Answer 57

jk

Answer 58

alright, I guess I am done with math for now

Answer 59

\(\huge \color{blue }{ت}\)

Answer 60

tnx
@ganeshie8
@SithsAndGiggles
my patience (:D)

marki, :PPPP

Answer 61

:P :P

Answer 62

\(\large T~~H~~A~~N~~K \small\displaystyle \int_{}^{} x~dx \)

Answer 63

got to go. excuse me
discúlpeme jk

Answer 64

ganeshie8... I am leaving... don't get too excited:P jk

Answer 65

I just got to go, but not for ever though, so sure.. go ahead...

Answer 66

\(\boxed{:O} \)

Answer 67

bye

Answer 68

ur funny :P bbye

Answer 69

\(\huge \color{red}{\ddot \smile} \)

Answer 70

huh my T in cuter :P