Probleme

Să se calculeze $l=\lim_{\alpha\to\infty}\displaystyle\int_0^{\alpha}\dfrac{2x+1}{x^4+2x^3+3x^2+2x+2}\,dx$.

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

Fie $f:(1,\infty)\to\mathbb{R}$, $f(y)=\displaystyle\int_0^y \dfrac{1}{x^2-2x+y}\,dx$. Calculați $\displaystyle\int_2^{10} f(y)\,dy$.

a) $3\pi$; b) $2\pi$; c) $\dfrac{5\pi}{3}$; d) $\pi$; e) $\dfrac{\pi}{2}$; f) $\dfrac{3\pi}{2}$.