Question

Integrala de la 0 la 1 din (x^2012+x^2011+x^2+x) totul supra x+1=?

Answer 1

$\displaystyle \frac{x^{2012}+x^{2011}+x^2+x}{x+1}= \frac{x^{2011}(x+1)+x(x+1)}{x+1}= \\ \\ =\frac{(x+1)(x^{2011}+x)}{x+1}=x^{2011}+x \\ \\ O~primitiva~a~lui~x^{2011}+x~este~functia~F(x)= \frac{x^{2012}}{2012}+ \frac{x^2}{2}. \\ \\ \int\limits^1_0 (x^{2011}+x) dx =F(1)-F(0)= \frac{1}{2012}+ \frac{1}{2}= \frac{1007}{2012}.$