$\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cdot \cos x} d x$ is equal to

$\int_0^2 x^3(2-x)^4 \, dx = $

The value of $\int_{4}^{7} \frac{(11-x)^{2}}{x^{2}+(11-x)^{2}} d x$ is

Let $f$ be a non-constant continuous function for all $x \geq 0$. Let $f$ satisfy the relation $f(x) f(a-x)=1$ for some $a \in R^{+}$. Then, $I=\int_{0}^{a} \frac{d x}{1+f(x)}$ is equal to

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

If $f(x) = \begin{cases} e^{\cos x}\sin x, & |x| \le 2 \\ 2, & \text{otherwise} \end{cases}$,then $\int_{-2}^{3} f(x) dx$ is equal to