$\int_{0}^{\pi} \cos^3 x \, dx = $

The number of continuous functions $f:[0,1] \rightarrow [0,1]$ such that $f(x) < x^2$ for all $x \in (0,1]$ and $\int_{0}^{1} f(x) dx = \frac{1}{3}$ is:

If $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$,then $\int_0^{\frac{\pi}{2}} \log \sec x \, dx = $

If $S_n = \int_0^{\frac{\pi}{2}} \frac{\sin((2n-1)x)}{\sin x} dx$ and $n$ is an integer,then $S_{n+1} - S_n =$

The value of $\int_{-2}^{2} (ax^3 + bx + c) dx$ depends on

Let $I_{n} = \int_{1}^{e} x^{19}(\log |x|)^{n} dx$,where $n \in N$. If $(20) I_{10} = \alpha I_{9} + \beta I_{8}$,for natural numbers $\alpha$ and $\beta$,then $\alpha - \beta$ is equal to ..... .