$\int_{-4}^{4} (2^x + 2^{-x})(3^x + 3^{-x}) \, dx$ is equal to

If ${I_n} = \int_{0}^{\pi /4} {\tan^n x} \,dx$,then $\lim_{n \to \infty} n[{I_n} + {I_{n - 2}}]$ equals

For any real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10, 10]$ by $f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$. Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos(\pi x) dx$ is:

If $I = \int_0^{\frac{\pi}{4}} \log (1 + \tan x) \, dx$,then the value of $I$ is

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cos^{\frac{3}{2}} x}{\cos^{\frac{3}{2}} x + \sin^{\frac{3}{2}} x} \, dx = $ . . . . . . .

$\int_5^9 \frac{\log_3 x^2}{\log_3 x^2 + \log_3(588 - 84x + 3x^2)} dx =$