$\int_{0}^{a} (a-x)^{\frac{3}{2}} x^{2} dx =$

$\int\limits_0^\pi {\frac{{x\cos x}}{{{{\left( {1 + \sin x} \right)}^2}}}} dx$ is equal to :

If $g(x) = \int_0^x \cos^4 t \,dt$, then $g(x+\pi)$ equals

By using the properties of definite integrals,evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x} dx$.

If $I = \int_0^\pi x \left\{ \sin^2(\sin x) + \cos^2(\cos x) \right\} dx$,then $[I] = \ldots$. Here,$[.]$ denotes the greatest integer function.

Let $f : R \rightarrow R$ be a continuous function satisfying $f(x) + f(x + k) = n$,for all $x \in R$,where $k > 0$ and $n$ is a positive integer. If $I_{1} = \int_{0}^{4nk} f(x) dx$ and $I_{2} = \int_{-k}^{3k} f(x) dx$,then: