Value of the definite integral $\int_{-1/2}^{1/2} (\sin^{-1}(3x - 4x^3) - \cos^{-1}(4x^3 - 3x)) dx$

$\int_{-a}^{a} \sqrt{\frac{a - x}{a + x}} dx =$

$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{1+e^x} d x=$

Let $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x}}{2}} \right)} dx = \alpha $. Then $\int\limits_0^1 {{{\tan }^{ - 1}}\left( {\frac{{\tan x - 2\cot x}}{3}} \right)} dx$ is equal to:

$\int_2^8 \frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}}+5^{\sqrt{10-x}}} d x$ is equal to

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+ e^{\sin x}} dx$ is: