Question

Se consideră funcţia $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=\left(x^{2}+2\right) e^{-x}$.

5p a) Arătați că $\int_{1}^{4} e^{x} f(x) d x=27$.

$5 p$ b) Calculați $\int_{1}^{e} f(\ln x) d x$.

$5 p$ c) Arătați că $\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} f(t) d t=2$.

Answer 1

$f(x)=\left(x^{2}+2\right) e^{-x}$

a)

Vezi tabelul de integrale din atasament

$\int\limits^4_1 {x^2+2} \, dx =\frac{x^3}{3}|_1^4+2x|_1^4=\frac{64}{3}-\frac{1}{3} +8-2=21+6=27$

b)

$\int\limits^e_1{\frac{ln^2x+2}{e^{lnx}} } \, dx =\int\limits^e_1\frac{ln^2x+2}{x} \ dx$

Desfacem in doua integrale si obtinem:
$\int\limits^e_1\frac{ln^2x}{x}\ dx+\int\limits^e_1\frac{2}{x} \ dx =\int\limits^e_1(lnx)'\cdot ln^2x\ dx+2lnx|_1^e=\frac{1}{3}ln^3x|_1^e +2lne-2ln1=\\\\=\frac{1}{3}ln^3e-\frac{1}{3}ln^31+2=\frac{1}{3} +2=\frac{7}{3}$

c)

$Stim \ ca \ \lim_{x \to a} \frac{F(x)-F(a)}{x-a} =f(a)$

Asadar, fie F primitiva lui f

F'(x)=f(x)

$F(x)=\int\limits {f(x)} \, dx$

$\int\limits^x_0 {f(t)} \, dx =F(t)|_0^x=F(x)-F(0)\\\\ \lim_{x \to 0} \frac{F(x)-F(0)}{x} =f(0)=2e^0=2$

Un alt exercitiu cu integrale gasesti aici: https://brainly.ro/tema/9919123

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