Question

Qual o resultado de ∫(2/x^3 +3/x^2 +5)dx?

Answer 1

Podemos reescrever a integral, assim:

∫$\frac{2}{ x^{3} }$ dx + ∫$\frac{3}{ x^{2} }$ dx + ∫5 dx


E ainda assim:

2 . ∫$\frac{1}{ x^{3} }$ dx + 3 . ∫$\frac{1}{ x^{2} }$ dx + ∫5 dx

2 . ∫$x^{-3}$ dx + 3 . ∫ $x^{-2}$ dx + ∫5 dx


Então, integramos:

2 . |$\frac{x^{-3 \ + \ 1}}{-3 \ + \ 1}$ + C|  + 3 . |$\frac{x^{-2 \ + \ 1}}{-2 \ + \ 1}$ + C|  + |5.x + C| 

2 . |$\frac{x^{-2}}{-2}$ + C| + 3 . |$\frac{x^{-1}}{-1}$ + C| + 5x + C

2 . ($\frac{x^{-2}}{-2}$) + 2.C + 3 . ($-x^{-1}$) + 3.C + 5x + C

$\frac{2.x^{-2}}{-2}$ + 2C + $-3.x^{-1}$ + 3C + 5x + C

$\frac{x^{-2}}{-1}$ + 2C + $-3x^{-1}$ + 3C + 5x + C


Resultado:

$-x^{-2}$ + $-3x^{-1}$ + 5x + 6C


Ou:

$\frac{1}{ x^{2} }$ + $\frac{-3}{x}$ + 5x + 6C