For $0 \le x \le \frac{\pi}{2}$,the value of $\int_{0}^{\sin^{2}x} \sin^{-1}(\sqrt{t}) \, dt + \int_{0}^{\cos^{2}x} \cos^{-1}(\sqrt{t}) \, dt$ is equal to

The number of elements in the set $S = \{x : x \in [0, 100] \text{ and } \int_{0}^{x} t^{2} \sin(x-t) dt = x^{2}\}$ is:

Let $[\bullet]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha\pi} \left( \frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta} \right) d\theta$ is equal to . . . . . . .

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

$\int_0^{2a} \frac{f(x)}{f(x) + f(2a - x)} \, dx = $

Let $T > 0$ be a fixed number. Suppose $f$ is a continuous function such that for all $x \in \mathbb{R}, f(x + T) = f(x)$. If $I = \int_{0}^{T} f(x) dx$,then the value of $\int_{3}^{3 + 3T} f(2x) dx$ is