Calculate the integral $\int e^{\sin x}\,dx$

Question

I found this exercise in a book so it probably has an elementary solution. I didn't succed but I hope maybe someone here will solve it.

$$\int e^{\sin x}\,dx$$

http://www.wolframalpha.com/input/?i=Integrate%5BExp%5BSin%5Bx%5D%5D,x%5D — parsiad, Apr 01 '17 at 14:40
u right but I dont knew what to say about it, so wird that it come from my teaching book of a course. — Yarden Sharabi, Apr 01 '17 at 14:43
yes and I'm seeing that it's realy difficult so I leaved it. — Yarden Sharabi, Apr 01 '17 at 14:59

score 4 · Answer 1 · answered Jun 23 '18 at 09:45

4

Elementary? Not much. One of the applicable methods explained in

While solving the integral of $e^{\sin(x)}$

$\displaystyle \int e^{\sin(x)} dx=I_0(1)x + \frac{\pi}{2}L_0(1) + 2\sum_{n=1}^{+\infty} \frac{I_n(1)}{n} \sin \left ( nx - \frac{n\pi}{2} \right ) $

$I_n(1)$ is rapidly diminishing and the shape of the integral is perfectly explained through this formula.

answered Jun 23 '18 at 09:45

Aleksandar Perisic

49

Wow. Had always been trying to figure what the Fourier expansion was for this with no luck. Imo this is the most elegant method of expressing this integral. – The_Sympathizer Sep 27 '19 at 09:33

Masacroso · Answer 2 · 2017-04-01T15:11:02.033

2

We have that

$$\int e^{\sin x}\mathrm dx=\int\left(\sum_{k=0}^\infty\frac{\sin^k(x)}{k!}\right)\mathrm dx=\sum_{k=0}^\infty\left(\int\frac{\sin^k(x)}{k!}\mathrm dx\right)=\sum_{k=0}^\infty\frac1{k!}\left(\int\sin^k(x)\mathrm dx\right)$$

where the swap between the summation and the integral is possible because the series converges uniformly for $x\in\Bbb R$. But I dont think that we can go far further than the last expression.

However, from the last expression, we can approach easily to any desired level of error the value of any definite integral.

We knows too that

$$I_n=\frac1n((n-1)I_{n-2}-\cos x(\sin x)^{n-1})$$

for $I_n:=\int\sin^n(x)\mathrm dx$. But as @MrYouMath said in the end we will need numerical methods, and we can use it directly in the original integral, by example using any kind of Simpson's rule.

edited Apr 01 '17 at 15:11

answered Apr 01 '17 at 14:55

Masacroso

30,417

You could also expand the sine function. – MrYouMath Apr 01 '17 at 15:00
@MrYouMath ya, I know but... And we can find a recurrence relation to write $\int \sin^n(x)\mathrm dx$ – Masacroso Apr 01 '17 at 15:01
1

But in the end, it is easier to calculate the integral directly by numerical methods :D. – MrYouMath Apr 01 '17 at 15:08
it cen be a nice exercise to show in that way that $\int_{0}^\infty{e^{\sin(x)}}<\infty$. thank u for that conclution. – Yarden Sharabi Apr 01 '17 at 15:26
@YardenSharabi if you graph the function you can see that $\int_0^\infty e^{\sin (x)}\mathrm dx=\infty$ – Masacroso Apr 01 '17 at 15:33
u right I did a mistake. Thanke u again! – Yarden Sharabi Apr 01 '17 at 15:43

score 2 · Answer 3 · answered Jun 23 '18 at 11:04

Copying my answer to the almost duplicate question

See also https://math.stackexchange.com/a/117545/442

Although this indefinite integral has no known closed form, certain definite integrals are known... $$ \int_{-\pi/2}^{\pi/2} e^{\sin x}\;dx = \pi\;I_0(1) , $$ where $I_0$ is a modified Bessel function

Calculate the integral $\int e^{\sin x}\,dx$

3 Answers3

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