Question

Va rog frumos ajutor la exercițiul din poza!0

Answer 1

(x³+x²+x+1)/(x²+1)=(x³+x)/(x²+1)+(x²+1)/(x²+1)=
x(x²+1)/(x²+1)+1=
x+1
Indegrala   devine
I=∫(x+1)dx=∫xdx+∫dx=x²/2+x+C

Answer 2

$\displaystyle \mathtt{\int\limits \frac{x^3+x^2+x+1}{x^2+1}dx }\\ \\ \mathtt{x^3+x^2+x+1=x^2(x+1)+(x+1)=(x+1)(x^2+1)}\\ \\ \mathtt{\int\limits \frac{x^3+x^2+x+1}{x^2+1}dx =\int\limits \frac{(x+1)(x^2+1)}{x^2+1}dx=\int\limits (x+1)dx=}\\ \\ \mathtt{=\int\limits xdx+\int\limits1dx= \frac{x^2}{2} +x+C}$