Let's do some calculus! (8)

Calculus Level 5

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

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

\(\begin{array} {} \quad & A: & a=2, & L=\dfrac{e^{4\pi}-1}{e^{\pi}-1} & \implies 2 \\ & B: & a=4, & L=\dfrac{e^{4\pi}+1}{e^{\pi}+1} & \implies 3 \\ & C: & a=2, & L=\dfrac{e^{4\pi}+1}{e^{\pi}+1} & \implies 5 \\ & D: & a=4, & L=\dfrac{e^{4\pi}-1}{e^{\pi}-1} & \implies 7 \end{array}\)

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


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