Question

Se consideră funcţia $f:(0,+\infty) \rightarrow \mathbb{R}, f(x)=\ln x$.

$5 \mathbf{p}$ a) Arătaţi că $\int_{1}^{e} f^{\prime}(x) d x=1$.

$5 \mathbf{p}$ b) Calculați $\int_{1}^{e} \frac{f^{2}(x)}{x} d x$.

$5 p$ c) Determinaţi numărul real $p, p\ \textgreater \ 1$, ştiind că $\int_{1}^{p} x f(x) d x=\frac{p^{2}}{2} \ln p-\frac{3}{4}$.

Answer 1

Răspuns:

a) $\displaystyle\int_1^pf'(x)dx=\left. f(x)\right|_1^e=f(e)-f(1)=1$

b) $\displaystyle\int_1^e\frac{f^(x)}{x}dx==\int_1^ef^2(x)f'(x)dx=\left. \frac{f^3(x)}{3}\right|_1^e=\frac{1}{3}$

c) $\displaystyle\int_1^pxf(x)dx=\int_1^p\left(\frac{x^2}{2}\right)'\ln xdx=\left.\frac{x^2}{2}\ln x\right|_1^p-\frac{1}{2}\int_1^pxdx=\frac{p^2}{2}\ln p-\frac{p^2}{4}+\frac{1}{4}$

Rezultă

$\displaystyle\frac{p^2}{2}\ln p-\frac{p^2}{4}+\frac{1}{4}=\frac{p^2}{2}\ln p-\frac{3}{4}\Rightarrow p^2=4\Rightarrow p=2$

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