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.

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

Answer 1

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.