For each positive integer $n$,define $f_n(x) = \min\left(\frac{x^n}{n!}, \frac{(1-x)^n}{n!}\right)$ for $0 \leq x \leq 1$. Let $I_n = \int_{0}^{1} f_n(x) dx$ for $n \geq 1$. Then,$\sum_{n=1}^{\infty} I_n$ is equal to

$\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x}+\sqrt[3]{\operatorname{cosec} x}} d x=$

If $f$ is a continuous function,then which of the following is true?

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:

$\int_{\log \frac{1}{2}}^{\log 2} \sin \left(\frac{e^x-1}{e^x+1}\right) d x=$

$\int_0^1 \frac{\log _e(1+x)}{1+x^2} d x=$