Integration of $e^{\cos x}\cos x$

Question

Could you please help me to solve this integration problem? I could not find an exact symbolic expression for it. $$\int {{e^{\cos x}}} \cos xdx$$

Answer 1

The indefinite integral knows no closed form expression, as can be proven by using either Liouville's theorem or the Risch algorithm. However, if we add $0$ and $\dfrac\pi2$ as the two limits of integration, then the result is $\dfrac\pi2\cdot\Big[I_1(1)+L_{-1}(1)\Big]$, where I and L are the Bessel and Struve functions, respectively.

Answer 2

As Lucian said there is no closed form expression for this undefinite integral, but the modified Bessel function of the first kind can be defined with a definite integral of the form (for integer $n$):

$$I_n(x)=\frac{1}{\pi}\int_0^{\pi}e^{x\cos u}\cos(nu)du$$

Answer 3

As said by Lucian, there is no closed form expression. As said by Lucian and TylerHG, there are quite nice expressions for the integral $$\begin{align} F_k&=\int_0^{k \frac{\pi}{2}}e^{\cos( x)}\cos(x)dx\\F_1&=\frac{1}{2} \pi ({L}_{-1}(1)+I_1(1))\\F_2&=\pi I_1(1)\\F_3&=\frac{1}{2} \pi (3 I_1(1)-{L}_{-1}(1))\\F_4&=2 \pi I_1(1)\\F_5&=\frac{1}{2} \pi ({L}_{-1}(1)+5 I_1(1))\\F_6&=3 \pi I_1(1)\\F_7&=\frac{1}{2} \pi (7 I_1(1)-{L}_{-1}(1))\\ F_8&=4 \pi I_1(1) \\F_9&=\frac{1}{2} \pi ({L}_{-1}(1)+9 I_1(1))\end{align}$$ where obvious patterns can be noticed.