Probleme

Valoarea limitei $\lim_{x\to 0}\dfrac{1}{x^4}\int_{x^2}^{2x^2}\dfrac{t}{\sqrt{t^2+9}}\,dt$ este:

a) $2$; b) $1$; c) $0$; d) $\infty$; e) $\dfrac{1}{2}$; f) $-1$.

Fie $f:\mathbb{R}\to\mathbb{R}$ o funcție derivabilă, astfel încât $x^2+2\int_0^x t\cdot f(t)\,dt+2=(x^2+1)\cdot f(x)+\ln 2$, $(\forall)\, x\in\mathbb{R}$. Atunci $\int_0^1 f(x)\,dx$ este:

a) $\dfrac{\pi}{2}$; b) $\pi$; c) $1$; d) $\dfrac{\pi}{3}$; e) $\dfrac{\pi}{4}$; f) $\dfrac{3\pi}{4}$.