Question

Calcule a integral abaixo:
$\int\limits {\frac{2 x^{3} - x^{2} -2}{ x^{2} }} \, dx$

Answer 1

$\displaystyle I=\int\dfrac{2x^3-x^2-2}{x^2}\,dx\\\\ I=\int\left(\dfrac{2x^3}{x^2}-\dfrac{x^2}{x^2}-\dfrac{2}{x^2}\right)\,dx\\\\ I=\int(2x-1-2x^{-2})\,dx\\\\ I=\int 2x\,dx-\int 1\,dx-\int 2x^{-2}\,dx\\\\ I=2\int x\,dx-\int \,dx-2\int x^{-2}\,dx\\\\ I=2\left[\dfrac{x^2}{2}\right]-[x]-2\left[\dfrac{x^{-2+1}}{-2+1}\right]+C\\\\ I=x^2-x-2\left[\dfrac{x^{-1}}{-1}\right]+C\\\\ I=x^2-x+2x^{-1}+C\\\\ I=x^2-x+\dfrac{2}{x}+C\\\\ \boxed{\int\dfrac{2x^3-x^2-2}{x^2}\,dx=x^2-x+\dfrac{2}{x}+C}$