$\int_0^1 {\log \sin \left( {\frac{\pi }{2}x} \right)} \,dx = $

Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $,$I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $,and $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $. Consider the following statements:
$I: I_1 < I_2$
$II: I_2 < I_3$
$III: I_1 = I_3$
Which of the following is (are) true?

The value of $\int_{0}^{\sin^2 x} \sin^{-1} \sqrt{t} \, dt + \int_{0}^{\cos^2 x} \cos^{-1} \sqrt{t} \, dt$ for $x \in (0, \pi/2)$ is:

Evaluate the definite integral: $\int_0^{2a} f(x) dx$

If $\int_{0}^{1} 4 \cot^{-1}(1-x+x^{2}) dx = a \tan^{-1}(2) - b \log_{e}(5)$, where $a, b \in N$, then $(2a+b)$ is equal to :

$\int_0^{\pi /2} {\frac{1}{{1 + \sqrt {\tan x} }}} \,dx = $