Definite integral of a cosine function times a complex exponential also of a cosine function

Question

I need to solve these integrals:

$$ \int_0^{2\pi}cos(4\theta) e^{I k \rho cos(\theta)}\text{d}\theta $$

and

$$ \int_0^{2\pi}sin^4(\theta) e^{I k \rho cos(\theta)}\text{d}\theta $$

I couldn't solve any of them on wolfram and I didn't find any table of integrals that could help-me

Answer 1

For $\int_0^{2\pi}e^{Ik\rho\cos\theta}\cos4\theta~d\theta$ ,

$\int_0^{2\pi}e^{Ik\rho\cos\theta}\cos4\theta~d\theta$

$=\int_0^\pi e^{Ik\rho\cos\theta}\cos4\theta~d\theta+\int_\pi^{2\pi}e^{Ik\rho\cos\theta}\cos4\theta~d\theta$

$=\int_0^\pi e^{Ik\rho\cos\theta}\cos4\theta~d\theta+\int_\pi^0e^{Ik\rho\cos(2\pi-\theta)}\cos(4(2\pi-\theta))~d(2\pi-\theta)$

$=\int_0^\pi e^{Ik\rho\cos\theta}\cos4\theta~d\theta+\int_0^\pi e^{Ik\rho\cos\theta}\cos4\theta~d\theta$

$=2\int_0^\pi e^{Ik\rho\cos\theta}\cos4\theta~d\theta$

$=2\pi I_4(Ik\rho)$ (according to http://people.math.sfu.ca/~cbm/aands/page_376.htm)

For $\int_0^{2\pi}e^{Ik\rho\cos\theta}\sin^4\theta~d\theta$ ,

$\int_0^{2\pi}e^{Ik\rho\cos\theta}\sin^4\theta~d\theta$

$=\int_0^\pi e^{Ik\rho\cos\theta}\sin^4\theta~d\theta+\int_\pi^{2\pi}e^{Ik\rho\cos\theta}\sin^4\theta~d\theta$

$=\int_0^\pi e^{Ik\rho\cos\theta}\sin^4\theta~d\theta+\int_\pi^0e^{Ik\rho\cos(2\pi-\theta)}\sin^4(2\pi-\theta)~d(2\pi-\theta)$

$=\int_0^\pi e^{Ik\rho\cos\theta}\sin^4\theta~d\theta+\int_0^\pi e^{Ik\rho\cos\theta}\sin^4\theta~d\theta$

$=2\int_0^\pi e^{Ik\rho\cos\theta}\sin^4\theta~d\theta$

$=\dfrac{6\pi I_2(Ik\rho)}{I^2k^2\rho^2}$ (according to http://people.math.sfu.ca/~cbm/aands/page_376.htm)