Question

Se considera functia f:R-R, f(x)=x/(x^2+1)
a.Calculati integrala din f(x)*f'(x)dx
b.Determinati limita sirului an=[1/(n^2+1^2)]+[2/(n^2+2^2)]+...+[n/(n^2+n^2)]

Answer 1

$\displaystyle a)~Se~stie~ca~(f^2)'=2ff'. \\ \\ Deci~ \int f(x)f'(x) dx =\int \left( \frac{f^2(x)}{2} \right)'dx= \frac{f^2(x)}{2}+C.$

$\displaystyle b)~Limita~ceruta~este: \\ \\ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{k}{n^2+k^2}=\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{\frac{k}{n^2}}{1+\frac{k^2}{n^2}}=\lim_{n \to \infty} \frac{1}{n} \sum\limits_{k=1}^n \frac{\frac{k}{n}}{1+ \left(\frac{k}{n}\right)^2}= \\ \\ =\lim_{n \to \infty} \frac{1}{n} \sum\limits_{k=1}^n f \left (\frac{k}{n} \right )= \int\limits_0^1 f(x)dx = \int\limits_0^1 \frac{1}{2} \cdot \frac{(x^2+1)'}{x^2+1} dx=$
$\displaystyle =\left. \frac{1}{2}\ln(x^2+1) \right |^1_0= \frac{\ln2}{2}.$