$\int_1^e \frac{1}{x} \, dx$ is equal to

If $[x]$ is the greatest integer function not greater than $x$,then $\int_0^8 [x] dx$ is equal to

The function $L(x) = \int_1^x \frac{dt}{t}$ satisfies the equation

$\int_{0}^{\frac{\pi}{2}} \frac{4x \sin x + x^2 \cos x}{2\sqrt{\sin x}} dx$ is equal to

Let $I = \int_a^b (x^4 - 2x^2) dx$. If $I$ is minimum,then the ordered pair $(a, b)$ is

Define a function $f: R \rightarrow R$ by $f(x) = \max \{|x|, |x-1|, \ldots, |x-2n|\}$,where $n$ is a fixed natural number. Then,$\int_0^{2n} f(x) dx$ is