Evaluate the definite integral $\int_{4}^{5} e^{x} \, dx$.

$\int_{a}^{b} \operatorname{sgn}(x) \, dx = \dots$ (where $a, b \in \mathbb{R}$)

Evaluate the definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x$.

If $\int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x + \csc x} dx = m(\pi + n)$,then $m \cdot n$ is equal to

The limit of the area under the curve $y = e^{-x}$ from $x = 0$ to $x = h$ as $h \rightarrow \infty$ is:

The number of continuous functions $f:[0,1] \rightarrow \mathbb{R}$ that satisfy $\int_0^1 x f(x) dx = \frac{1}{3} + \frac{1}{4} \int_0^1 (f(x))^2 dx$ is