$I = \int\limits_0^1 \sqrt[3]{2x^3 - 3x^2 - x + 1} \, dx$ is equal to

If $\int_{-1}^4 f(x) dx = 4$ and $\int_2^4 (3 - f(x)) dx = 7$,then $\int_{-1}^2 f(x) dx = $

If $f(t) = \int_0^\pi \frac{2x \, dx}{1 - \cos^2 t \sin^2 x}$,where $0 < t < \pi$,then the value of $\int_0^{\frac{\pi}{2}} \frac{\pi^2 \, dt}{f(t)}$ equals..........

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:

By using the properties of definite integrals,evaluate the integral $\int_{0}^{2 \pi} \cos ^{5} x \, dx$.

$ \int_{0}^{\pi / 4} \log \left(\frac{\sin x+\cos x}{\cos x}\right) d x $