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

$\int_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x$ is equal to.

$\int_5^9 \frac{\log_3 x^2}{\log_3 x^2 + \log_3(588 - 84x + 3x^2)} dx =$

$\int_{0}^{\pi} \frac{\cos ^{4} x}{\cos ^{4} x+\sin ^{4} x} d x$ is equal to

The value of the integral $\int_{-\pi / 2}^{\pi / 2} \frac{\sin^2 x}{1+e^x} \, dx$ is

Let $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$ for all $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the correct expression$(s)$ is(are):
$(A) \int_0^{\pi/4} x f(x) dx = \frac{1}{12}$
$(B) \int_0^{\pi/4} f(x) dx = 0$
$(C) \int_0^{\pi/4} x f(x) dx = \frac{1}{6}$
$(D) \int_0^{\pi/4} f(x) dx = 1$