Let $${I_1} = \int_0^{4\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} $$

$$ = \int_0^\pi {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt + \int_\pi ^{2\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt + \int_{2\pi }^{3\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt + \int_{3\pi }^{4\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} } } } $$

$$\therefore$$ $${I_1} = {I_2} + {I_3} + {I_4} + {I_5}$$ ...... (i)

Now, $${I_3} = \int_\pi ^{2\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} $$

Put $$t = \pi + t \Rightarrow dt = dt$$

$$\therefore$$ $${I_3} = \int_0^\pi {{e^{\pi + t}}.\,({{\sin }^6}at + {{\cos }^6}at)dt} $$

$$ = {e^t}\,.\,{I_2}$$ ..... (ii)

Now, $${I_4} = \int_{2\pi }^{3\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} $$

Put $$t = 2\pi + t \Rightarrow dt = dt$$

$$\therefore$$ $${I_4} = \int_0^\pi {{e^{t + 2\pi }}({{\sin }^6}at + {{\cos }^6}at)dt} $$

$$ = {e^{2\pi }}.{I_2}$$ ...... (iii)

and $${I_5} = \int_{3\pi }^{4\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} $$

Put $$t = 3\pi + t$$

$$\therefore$$ $${I_5} = \int_0^\pi {{e^{3\pi + t}}({{\sin }^6}at + {{\cos }^6}at)dt} $$

$$ = {e^{3\pi }}.\,{I_2}$$ ...... (iv)

From Eqs. (i), (ii), (iii) and (iv), we get

$${I_1} = {I_2} + {e^\pi }.\,{I_2} + {e^{2\pi }}.\,{I_2} + {e^{3\pi }}.\,{I_2}$$

$$ = (1 + {e^\pi } + {e^{2\pi }} + {e^{3\pi }}){I_2}$$

$$\therefore$$ $$L = {{\int_0^{4\pi } {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} } \over {\int_0^\pi {{e^t}({{\sin }^6}at + {{\cos }^6}at)dt} }}$$

$$ = (1 + {e^\pi } + {e^{2\pi }} + {e^{3\pi }})$$

$$ = {{1.\,({e^{4\pi }} - 1)} \over {{e^\pi } - 1}}$$ for $$a \in R$$