$\int_0^{2 \pi} \sin ^3 x \cos ^2 x \, dx = $ . . . . . . .

The value of $\int_{-2}^{2} (ax^3 + bx + c) dx$ depends on

The value of $\int_1^3 \sqrt{3 + x^3} \,dx$ lies in the interval

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\tan^4 x} dx = $ . . . . . . .

Let $f(x)$ be a positive function,$I_1 = \int_{-\frac{1}{2}}^1 2x f(2x(1-2x)) dx$,and $I_2 = \int_{-1}^2 f(x(1-x)) dx$. Then the value of $\frac{I_2}{I_1}$ is equal to:

$\int_{a-6}^{b-6} f(x+6) dx$ is equal to