Question

se considera functia f:R R , f(x)= x^5+x^3+2x. aratati ca $\int\limits^2_0 {e^x(f(x)-x^5-x^3+1)} \, dx =3e^2+1$

Answer 1

[tex]f : \mathbb_{R} \rightarrow \mathbb_{R}, \quad $ $ f(x) = x^5+x^3+2x \\ \\ \int\limits^2_0 e^x\Big(f(x)-x^5-x^3+1\Big)} \, dx =\\ \\ = \int\limits^2_0 e^x\Big(x^5+x^3+2x-x^5-x^3+1\Big)} \, dx = \int\limits^2_0 e^x\cdot \big(2x+1\big)} \, dx= \\ \\ = \int\limits^2_0 (e^x)'\cdot \big(2x+1\big)} \, dx=e^x\cdot (2x+1)\Big|_0^2-\int\limits^2_0 e^x\cdot \big(2x+1\big)}' \, dx = \\ \\ = e^2\cdot (2\cdot 2+1) - e^0\cdot (2\cdot 0+1) -\int\limits^2_0 e^x\cdot 2 \, dx = \\ \\ [/tex]

$=e^2\cdot 5-1-2\cdot \int\limits^2_0 e^x = 5e^2-1-2\cdot (e^x)\Big|_0^2 = \\ \\ = 5e^2-1-2\cdot (e^2-e^0) = 5e^2-1-2\cdot (e^2-1) = 5e^2-1-2e^2+2 = \\ \\ = 3e^2+1$