Subiectul III, ex.2.a.

Se consideră funcţia \(f:(0,+\infty)\rightarrow (0,+\infty),\;\;f(x)=x+\frac{9}{x}\).
Arătaţi că \(\int_1^3\left( f(x)-\frac{9}{x}\right)dx=4\).
Soluţie. \begin{align} &\cssId{Step1}{\int_1^3\left( f(x)-\frac{9}{x}\right)dx=}\\ &\cssId{Step2}{=\int_1^3\left( x+\frac{9}{x}-\frac{9}{x}\right)dx=}\\ &\cssId{Step3}{=\int_1^3xdx=\frac{x^2}{2}\mid_1^3=}\\ &\cssId{Step4}{=\frac{3^2}{2}-\frac{1^2}{2}=}\\ &\cssId{Step5}{=\frac{9}{2}-\frac{1}{2}=\frac{8}{2}=4}\\ \end{align}