QJEEAdv15Ma9268: JEE Advanced 2015 Mathematics - Class 12 JEEAdv15Ma9268 Solution: The option(s) with the values of a and that satisfy the…

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 $a \in R$