$\int_{-1}^1 \sin^5 x \cos^4 x \, dx = $ . . . . . .

$\int_0^\pi \frac{x \tan x}{\sec x \cdot \operatorname{cosec} x} d x$ is equal to

On the real line $R$,we define two functions $f$ and $g$ as follows:
$f(x) = \min \{x - [x], 1 - x + [x]\}$
$g(x) = \max \{x - [x], 1 - x + [x]\}$
where $[x]$ denotes the greatest integer function. The positive integer $n$ for which $\int_0^n (g(x) - f(x)) \, dx = 100$ is:

Let $f: [2, 5] \to [2, 5]$ be a bijective function such that $\frac{d}{dx}(f^{-1}(x)) > 0$ for all $x \in [2, 5]$. Then $\int_{2}^{5} (f(x) + f^{-1}(x)) dx$ is

If $[ \cdot ]$ represents the greatest integer function,then $\int_{-1}^1 (x[1+\sin(\pi x)]+1) dx = $

$\int_{\frac{\pi}{3}}^{\frac{2 \pi}{3}} \frac{x}{1+\sin x} \,d x=$