The approximate value of $\int_{1}^{9} x^2 dx$ by using the trapezoidal rule with $4$ equal intervals is:

Let $\alpha \in (0,1)$ and $\beta = \log_{e}(1-\alpha)$. Let $P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \dots + \frac{x^n}{n}$ for $x \in (0,1)$. Then the integral $\int_{0}^{\alpha} \frac{t^{50}}{1-t} dt$ is equal to

Let $[x]$ denote the greatest integer less than or equal to $x$, then the value of the integral $\int_{-1}^{1}(|x|-2[x]) \, dx$ is equal to

Evaluate the definite integral $\int_{-2}^{2.24} [x] \, dx$,where $[x]$ denotes the greatest integer function.

Evaluate the definite integral $\int_2^5 2[x] \, dx$,where $[x]$ denotes the greatest integer function $\leq x$.

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