The value of the integral $\int_{0}^{\pi / 2} (\sin^{100} x - \cos^{100} x) dx$ is

Let $T > 0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T) = f(x)$ for all $x \in R$. If $I = \int_0^T f(x) dx$,then find the value of $\int_0^{5T} f(2x) dx$.

Let $f$ be a positive function. Let $I_1 = \int_{1 - k}^k x f\{x(1 - x)\} dx$ and $I_2 = \int_{1 - k}^k f\{x(1 - x)\} dx$,where $2k - 1 > 0$. Then $I_1/I_2$ is

If $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$,then:

$\int_0^{1000} e^{x-[x]} \, dx$ is equal to

$\int_{0}^{2a} f(x) \, dx = $