$\int_{0}^{\pi} \sin x \, dx = $ . . . . . . .

If $\int_{0}^{1}(5x^{2}-3x+k)dx=0$,then $k=$

If $\int_{n}^{n+1} f(x) dx = n^2 + n$ for all $n \in I$,then the value of $\int_{-3}^{3} f(x) dx$ is equal to

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin (x-[x]) \, dx=$

$\int_0^{\frac{\pi}{4}} \sec^4 x \, dx =$

Let $f : [1/2, 1] \rightarrow \mathbb{R}$ be a positive,non-constant and differentiable function such that $f^{\prime}(x) < 2f(x)$ and $f(1/2) = 1$. Then the value of $\int_{1/2}^1 f(x) dx$ lies in the interval