$\int_2^4 \frac{\log x^2}{\log x^2+\log (36-12x+x^2)} dx$ is equal to

Consider $L = \sqrt[3]{2012} + \sqrt[3]{2013} + \ldots + \sqrt[3]{3011}$,$R = \sqrt[3]{2013} + \sqrt[3]{2014} + \ldots + \sqrt[3]{3012}$,and $I = \int_{2012}^{3012} \sqrt[3]{x} \, dx$. Then,

$\int\limits_a^b [x] \,dx + \int\limits_a^b [-x] \,dx$,where $[.]$ denotes the greatest integer function,is equal to:

If $f(a+b+1-x)=f(x)$ for all $x,$ where $a$ and $b$ are fixed positive real numbers,then $\frac{1}{a+b} \int_{a}^{b} x(f(x)+f(x+1)) dx$ is equal to:

The value of $\int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6} dx$ is

If $f(x) = \frac{e^x}{1 + e^x}$,$I_1 = \int_{f(-a)}^{f(a)} x g\{x(1 - x)\} dx$,and $I_2 = \int_{f(-a)}^{f(a)} g\{x(1 - x)\} dx$,then the value of $\frac{I_2}{I_1}$ is