Question

se considera functia f:R→R , x^5 +x^3+2x. aratati ca $se \int\limits^1_1 {(f(x)-x^3-2x )} \, dx =0$

Answer 1

$f : \mathbb_{R} \rightarrow \mathbb_{R},\quad f(x) = x^5+x^3+2x\\ \\ \\ \int\limits^{1}_{-1} \Big({f(x) - x^3-2x} \Big)\, dx = \int\limits^{1}_{-1} \Big({x^5+x^3+2x - x^3-2x} \Big)\, dx= \\ \\ =\int\limits^{1}_{-1}x^5\, dx = \dfrac{x^{5+1}}{5+1}\Big|_{-1}^1= \dfrac{x^6}{6}\Big|_{-1}^1 =\dfrac{1^6}{6} - \dfrac{(-1)^6}{6} = \dfrac{1}{6}-\dfrac{1}{6} = 0$