Question

4)Cea de a4a întrebare
Mulțumesc !

Answer 1

 

$\displaystyle\bf\\Se da:\\\\f:R\to R,~f(x)=\frac{x}{x^2+1}\\\\Se~cere~sa~aratam~ca:\\\\\int\limits^1_{-1} f(x)\cdot (x^2+1)~dx=0\\\\\\ Rezolvare:\\\\\int\limits^1_{-1} f(x)\cdot (x^2+1)~dx=\int\limits^1_{-1} \frac{x}{x^2+1}\cdot (x^2+1)~dx=\\\\\\=\int\limits^1_{-1} \frac{x(x^2+1)}{x^2+1}~dx=~~(Simplificam~cu~(x^2+1))\\\\\\=\int\limits^1_{-1} x~dx=\left\frac{x^2}{2}\right |_{-1}^1=\frac{1^2}{2}-\frac{(-1)^2}{2}= \frac{1}{2}-\frac{1}{2}=0\\\\\\cctd$