Răspuns:
a) [tex]\displaystyle\int_1^pf'(x)dx=\left. f(x)\right|_1^e=f(e)-f(1)=1[/tex]
b) [tex]\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}[/tex]
c) [tex]\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}[/tex]
Rezultă
[tex]\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[/tex]
Explicație pas cu pas:
