Question

A integral de .......

Answer 1

$\displaystyle\int{\sqrt{x}\left(x+\dfrac{1}{x} \right )\,dx}\\ \\ \\ =\int{x^{1/2}\cdot \left(x+x^{-1} \right )\,dx}\\ \\ \\ =\int{(x^{1/2}\cdot x+x^{1/2}\cdot x^{-1})\,dx}\\ \\ \\ =\int{(x^{(1/2)+1}+x^{(1/2)-1})\,dx}\\ \\ \\ =\int{(x^{3/2}+x^{-1/2})\,dx}\\ \\ \\ =\dfrac{x^{(3/2)+1}}{\frac{3}{2}+1}+\dfrac{x^{(-1/2)+1}}{(-\frac{1}{2})+1}+C\\ \\ \\ =\dfrac{x^{5/2}}{(\frac{5}{2})}+\dfrac{x^{1/2}}{(\frac{1}{2})}+C\\ \\ \\ =\dfrac{2}{5}\,x^{5/2}+2x^{1/2}+C\\ \\ \\ =\dfrac{2}{5}\,\sqrt{x^{5}}+2\sqrt{x}+C$