One of the possible functions $f(x)$ which satisfies $\int_{-2}^{2} f(x) dx = 0$ is

The value of $I=\int_{\sqrt{\log _e 2}}^{\sqrt{\log _e 3}} \frac{x \sin x^2}{\sin x^2+\sin \left(\log _e 6-x^2\right)} d x$ is

If $P = \int_0^{3\pi} f(\cos^2 x) dx$ and $Q = \int_0^{\pi} f(\cos^2 x) dx$,then:

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

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

The value of the integral $\int_{-2}^{2} \frac{\sin^2 x}{[\frac{x}{\pi}] + \frac{1}{2}} \, dx$ (where $[x]$ denotes the greatest integer less than or equal to $x$) is