Let $$f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$$ for all $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$$
Then the correct expression(s) is (are)

The option(s) with the values of a and $$L$$ that satisfy the following equation is (are) $$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} }} = L?$$$

Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right) $$= $$\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt$$. Then

Let a $$\in$$ R and f : R $$\to$$ R be given by f(x) = x⁵ $$-$$ 5x + a. Then,