If $\int_1^n [x] dx = 120$,then $n = $

If $I = \int_0^{100\pi} \sqrt{1 - \cos 2x} \, dx$,then the value of $I$ is

The value of the definite integral $\int\limits_0^{\frac{1}{{\sqrt 2 }}} {\frac{{{x^2}dx}}{{\sqrt {1 - {x^2}} \,(1 + \sqrt {1 - {x^2}} )}}} $ is

Evaluate the definite integral: $\int_{0}^{1} \left(1 - \frac{x}{1!} + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots \infty\right) e^{2x} \, dx$.

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:

$\int_0^1 x^{5/2} (1-x)^{3/2} \, dx =$