Integral Exponential Cosine 827A94

1. **State the problem:** We want to find the integral $$\int e^{3\cos(2x)} \, dx$$. 2. **Formula and approach:** This integral involves an exponential function with a trigonometric argument. There is no elementary antiderivative for this integral in terms of elementary functions. 3. **Rewrite the integral:** We can express $$e^{3\cos(2x)}$$ using the series expansion or special functions, but typically this integral is expressed in terms of the modified Bessel function of the first kind, $$I_n(z)$$. 4. **Using the expansion:** Recall the identity: $$e^{z \cos \theta} = I_0(z) + 2 \sum_{n=1}^\infty I_n(z) \cos(n \theta)$$ 5. **Apply to our integral:** Here, $$z=3$$ and $$\theta=2x$$, so $$e^{3\cos(2x)} = I_0(3) + 2 \sum_{n=1}^\infty I_n(3) \cos(2nx)$$ 6. **Integrate term-by-term:** $$\int e^{3\cos(2x)} dx = \int I_0(3) dx + 2 \sum_{n=1}^\infty I_n(3) \int \cos(2nx) dx$$ 7. **Integrate cosine terms:** $$\int \cos(2nx) dx = \frac{\sin(2nx)}{2n} + C$$ 8. **Final expression:** $$\int e^{3\cos(2x)} dx = I_0(3) x + \sum_{n=1}^\infty \frac{I_n(3)}{n} \sin(2nx) + C$$ **Summary:** The integral does not have a simple elementary form but can be expressed as an infinite series involving modified Bessel functions of the first kind. **Answer:** $$\boxed{\int e^{3\cos(2x)} dx = I_0(3) x + \sum_{n=1}^\infty \frac{I_n(3)}{n} \sin(2nx) + C}$$