Question

Exercitiul 2 punctul a si c

Answer 1

$\displaystyle \mathtt{2.~~~a)~\int\limits_0^1 \frac{1}{e^x}f(x)dx= \frac{5}{2}}\\ \\ \mathtt{\int\limits_0^1 \frac{1}{e^x}\cdot (3x+1)e^xdx=\int\limits_0^1(3x+1)dx=\int\limits_0^13xdx+\int\limits_0^11dx=}\\ \\ \mathtt{=3\int\limits_0^1xdx+\int\limits 1dx=3 \frac{x^2}{2}\Bigg|_0^1 +x\Bigg|_0^1=3 \left( \frac{1^2}{2}- \frac{0^2}{0}\right)+1-0=}\\ \\ \mathtt{=3 \cdot \frac{1}{2}+1= \frac{3}{2}+1= \frac{3+2}{2}= \boxed{\mathtt{\frac{5}{2}}} }$

$\displaystyle \mathtt{c)~\int\limits_0^af(x)dx=3a}\\ \\ \mathtt{\int\limits_0^a(3x+1)e^xdx}\\ \\ \mathtt{\int\limits f(x)g'(x)dx=f(x)g(x)-\int\limits f'(x)g(x)dx}\\ \\ \mathtt{f(x)=3x+1\Rightarrow f'(x)=(3x+1)'=(3x)'+1'=3x'=3 \cdot 1=3}\\ \\ \mathtt{g'(x)=e^x\Rightarrow g(x)=\int\limits e^xdx=e^x}\\ \\ \mathtt{\int\limits (3x+1)e^xdx=(3x+1)e^x-\int\limits3e^xdx=(3x+1)e^x-3\int\limits e^xdx=}\\ \\ \mathtt{=(3x+1)e^x-3e^x+C=e^x(3x-2)+C}$

$\displaystyle \mathtt{\int\limits_0^a (3x+1)e^xdx=e^x(3x-2)\Bigg|_0^a=e^a(3a-2)-e^0(3 \cdot 0-2)=}\\ \\ \mathtt{=e^a(3a-2)-1\cdot (-2)=e^a(3a-2)+2}\\ \\ \mathtt{\int\limits_0^a(3x+1)e^xdx=3a \Rightarrow e^a(3a-2)+2=3a \Rightarrow} \\ \\ \mathtt{\Rightarrow 3ae^a-2e^a+2-3a=0\Rightarrow 3a\left(e^a-1\right)-2\left(e^a-1\right)=0\Rightarrow }\\ \\ \mathtt{\Rightarrow \left(e^a-1\right)(3a-2)=0\Rightarrow \boxed{\mathtt{a= \frac{2}{3}}} }$