$\int_{-\pi / 4}^{\pi / 4} x^3 \sin ^4(x) d x=$

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

The value of the integral $\int_0^{\pi / 2} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$ is

If $[t]$ denotes the greatest integer $\leq t$,then the value of $\frac{3(e-1)^2}{e} \int \limits_1^2 x^2 e^{[x]+[x^3]} dx$ is:

If $F(x) = f(x) + f\left(\frac{1}{x}\right)$,where $f(x) = \int_{1}^{x} \frac{\log_{e} t}{1+t} dt$,then $F(e) = $

$\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cdot \cos x} d x$ is equal to