Let $f(x)$ be positive for all real $x$. If $I_1 = \int_{1-h}^{h} x f(x(1-x)) dx$ and $I_2 = \int_{1-h}^{h} f(x(1-x)) dx$,where $(2h-1) > 0$,then $\frac{I_1}{I_2}$ is

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

The value of $\int_{-2}^{2} \left[ p \ln \left( \frac{1+x}{1-x} \right) + q \ln \left( \frac{1-x}{1+x} \right)^{-2} + r \right] dx$ depends on:

If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{-1}$ are two roots of the equation $x^3 + ax^2 + bx + c = 0$ with $a, b, c \in R$, then $\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx$ is equal to:

By using the properties of definite integrals,evaluate the integral $\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x$.

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