If $\displaystyle\int\dfrac{x^3\,dx}{\sqrt{1+x^2}} = a(1+x^2)^{3/2}+b\sqrt{1+x^2}+c$, where 'C' is an arbitrary constant, then the value of $a$ and $b$ are

$\int \frac{x^8}{(x^3+1)^{1/3}} dx$ is equal to

The value of $\int \frac{dx}{1+\sqrt{x}}$ is

If $2f(x) + f\left(\frac{1}{x}\right) = x^2 + 1$, then $\int f(x) dx$ is: (Here C is an arbitrary constant)

If $\int(\sqrt{x+1} + \sqrt{x-1})^2 dx = \alpha x^2 + \beta x\sqrt{x^2-1} + \gamma \log|x+\sqrt{x^2-1}| + C$, then value of $\alpha + \beta - 2\gamma$ is :

$\int_{\pi/6}^{\pi/3} \frac{tan x}{tan x + cot x} dx$ is equal to

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$, then $\int_{-1}^{1} f(x) \, dx =$