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.


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