Evaluate the integral: $\int_{-1}^{1} \left[ \sqrt{1+x+x^{2}} - \sqrt{1-x+x^{2}} \right] dx$

The value of $\int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ is

$\int_{-1}^{1} x^{17} \cos^{4} x \, dx = $

Which of the following statements is incorrect for the function $g(\alpha)$ for $\alpha \in R$ such that $g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin^{\alpha} x}{\cos^{\alpha} x+\sin^{\alpha} x} dx$?

Let $g(t) = \int_{-\pi/2}^{\pi/2} \cos \left(\frac{\pi}{4} t + f(x)\right) \, dx$,where $f(x) = \log_e \left(x + \sqrt{x^2 + 1}\right)$,$x \in R$. Then which one of the following is correct?

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: