JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Definite Integrals

The function $f(t)={\sin }^6(at)+{\cos }^4(at)$ is periodic with period $\frac{\pi }{a}$ (for integer $a$). For $a=2$, the period is $\frac{\pi }{2}$, so $4\pi$ contains 8 periods and $\pi$ contains 2 periods. For $a=4$, the period is $\frac{\pi }{4}$, so $4\pi$ contains 16 periods and $\pi$ contains 4 periods. Using the property of periodic functions in integrals, ${\int }_0^{n\pi /a}{e}^{t}f(t)dt$ can be expressed in terms of ${\int }_0^{\pi /a}{e}^{t}f(t)dt$. For both $a=2$ and $a=4$, the ratio simplifies to $L=\frac{e^{4\pi }-1}{e^{\pi }-1}$, confirming options (A) and (C).