Question

Se considera functiile f.F : R=>R date prin 
f(x)=$e^{x} + 3 x^{2} + 2 si F(x) = e^{x}+ x^{3} +2x-1$

c) Sa se demonstreze ca $\int\limits^1_0 {(xf(x)+F(x)} \, dx = F(1)$

Multumesc

Answer 1

Se observă că $F'(x)=f(x)$.
Atunci $\displaystyle\int_0^1\left[xf(x)+F(x)\right]dx=\int_0^1\left[xF'(x)+x'F(x)\right]dx=\\=\displaystyle\int_0^1\left[x\cdot F(x)\right]'dx=\left. x\cdot F(x)\right|_0^1=F(1)$