If $[x]$ is the greatest integer function not greater than $x$,then $\int_0^8 [x] dx$ is equal to

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in $A$.$P$. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}$ is:

$ \int_{0}^{\frac{1}{2}} \frac{dx}{(1+x^{2}) \sqrt{1-x^{2}}} $ is equal to

$\int_0^{\pi /8} \cos^3(4\theta) \, d\theta = $

If ${I_1} = \int_0^1 {2^{x^2}} dx$,${I_2} = \int_0^1 {2^{x^3}} dx$,${I_3} = \int_1^2 {2^{x^2}} dx$,and ${I_4} = \int_1^2 {2^{x^3}} dx$,then which of the following is true?

The value of the integral $\int_{\frac{1}{3}}^{1} \frac{\left(x-x^{3}\right)^{\frac{1}{3}}}{x^{4}} d x$ is