Question

Ajutor,va rog frumos

Answer 1

$\displaystyle \mathtt{\int\limits \left(x+ \frac{2}{x}\right)^2dx=\int\limits\left(x^2+2 \cdot x \cdot \frac{2}{x} +\left( \frac{2}{x} \right)^2\right)dx= }\\ \\ \mathtt{=\int\limits\left(x^2+ \frac{4x}{x} + \frac{2^2}{x^2} \right)dx=\int\limits\left(x^2+4+ \frac{4}{x^2}\right)dx= }$
$\displaystyle \mathtt{=\int\limits x^2dx+\int\limits4dx+\int\limits \frac{4}{x^2} dx=\int\limits x^2dx+4\cdot\int\limits1dx+4\cdot\int\limits \frac{1}{x^2}= }\\ \\ \mathtt{=\int\limits x^2dx+4\cdot\int\limits1dx+4 \cdot \int\limits x^{-2}dx= \frac{x^3}{3}+4x+4 \cdot \frac{x^{-1}}{-1}+C= }\\ \\ \mathtt{= \frac{x^3}{3}+4x+4 \cdot \left(- \frac{1}{x} \right)+C= \frac{x^3}{3}+4x- \frac{4}{x}+C }$