Let's do some calculus! (8)

Calculus Level 4

$\dfrac{\displaystyle \int_{0}^{4\pi}{e^{t}\left(\sin^{6}at + \cos^{4}at\right)} \,dt}{\displaystyle \int_{0}^{\pi}{e^{t}\left(\sin^{6}at + \cos^{4}at\right)} \,dt} = L$

Give your answer as the product of the numbers corresponding to the correct set of values of $$a$$ and $$L$$.

A) $$a=2$$ , $$L=\dfrac{e^{4\pi}-1}{e^{\pi}-1} \cdots (2)$$

B) $$a=4$$ , $$L=\dfrac{e^{4\pi}+1}{e^{\pi}+1} \cdots (3)$$

C) $$a=2$$ , $$L=\dfrac{e^{4\pi}+1}{e^{\pi}+1} \cdots (5)$$

D) $$a=4$$ , $$L=\dfrac{e^{4\pi}-1}{e^{\pi}-1} \cdots (7)$$

Notation: $$e \approx 2.71828$$ denotes the Euler's number.