$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}(x^5-x^3 \cos x+\sin^3 x-3) \, dx = $ . . . . . .

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

If $f(x) = \cos(\tan^{-1}x)$,then the value of the integral $\int_{0}^{1} x f''(x) dx$ is

If $I = \int_0^{\pi /4} \sin^2 x \, dx$ and $J = \int_0^{\pi /4} \cos^2 x \, dx$,then $I = $

The value of $\int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6} dx$ is

Let $f(x) = \frac{x}{(1+x^n)^{1/n}}$,$x \in R - \{-1\}$,$n \in N$,$n > 2$. If $f^n(x) = (f \circ f \circ f \dots \text{upto } n \text{ times})(x)$,then $\lim_{n \to \infty} \int_0^1 x^{n-2} (f^n(x)) dx$ is equal to $...............$.