Question

Subiectul ||| 2.b)
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Answer 1

$\displaystyle \mathtt{f:\mathbb{R}\rightarrow \mathbb{R},~f(x)=x^5+x^3+2x}\\ \\ \mathtt{ \int\limits^2_0 e^x\left(f(x)-x^5-x^3+1\right)dx=3e^2+1 }\\ \\ \mathtt{\int\limits^2_0e^x\left(x^5+x^3+2x-x^5-x^3+1\right)dx=\int\limits^2_0(2x+1)e^xdx}\\ \\ \mathtt{\int\limits f(x)g'(x)dx=f(x)g(x)-\int\limits f'(x)g(x)dx}\\ \\ \mathtt{f(x)=(2x+1)\Rightarrow f'(x)=2}\\ \\ \mathtt{g'(x)=e^x\Rightarrow g(x)=\int\limits g'(x)dx=\int\limits e^xdx=e^x}$
$\displaystyle \mathtt{\int\limits(2x+1)e^xdx=(2x+1)e^x-\int\limits 2e^xdx=(2x+1)e^x-2\int\limits e^xdx=}\\ \\ \mathtt{=(2x+1)e^x-2e^x+C=e^x(2x-1)+C}\\ \\ \mathtt{\int\limits^2_0(2x+1)e^xdx=e^x(2x-1)\Bigg|^2_0=e^2(2 \cdot 2-1)-e^0(2 \cdot 0-1)=}\\ \\ \mathtt{=e^2(4-1)-1(0-1)=e^2 \cdot 3-1 \cdot (-1)=\underline{3e^2+1}}$