Question

me ajudem integração de Fraçoes parciais.
5X²+20X+6/x³+2X²+x

Answer 1

Bom dia Alepluto!

Solução!

$\displaystyle\int \frac{5 x^{2} +20x+6}{ x^{3} 3+2x+x}$

Veja que o grau do expoente do denominador é maior,vamos fatorar.

$\displaystyle\int \frac{5 x^{2} +20x+6}{ x^{3} +2 x^{2} +x}=\displaystyle\int \frac{5 x^{2} +20x +6}{ x(x^{2} +2x+1)}=\displaystyle\int \frac{5 x^{2} +20x+6}{(x)(x+1)(x+1)} }$

Fazendo!

$\dfrac{A}{x}+ \dfrac{B}{(x+1)}+ \dfrac{C}{(x+1)^{2} }= \dfrac{[A(x+1)^{2}+B(x+1).x+x] }{x(x+1)^{2} }$


$\dfrac{[A(x+1)^{2}+B(x+1).x+x] }{x(x+1)^{2} } =\dfrac{5 x^{2} +20x +6}{ x(x^{2})} \\\\ \dfrac{[A(x+1)^{2}+B(x+1).x+C.x] }{x(x+1)^{2} } =\dfrac{5 x^{2} +20x +6}{ x(x+1)^{2}} \\\\\\ A(x+1)^{2}+B(x+1).x+C.x=5 x^{2} +20x +6\\\\\\ Sendo ~~x=1\\\\ -9=A(0)+B(0).-1+C(-1)\\\\\\ C=9\\\\\\\ Sendo~~ x=0\\\\\\ A=6\\\\\\ Sendo x=1\\\\\\ 31=4A+2.B+C\\\\\ 31=4.6+2.B+9\\\\\ 31=24+2B+9\\\\\\ 31-24-9=2B\\\\\\\ 2B=-2\\\\\ B= \dfrac{-2}{2}\\\\\\ B=-1$

Retomando a integral

$\displaystyle\int \frac{5 x^{2} +20x+6}{ x^{3} 3+2x+x}= \frac{6}{x}+ \frac{-1}{x+1} + \frac{9}{x+1}\\\\\\\\ \displaystyle\int \frac{A}{x}+ \frac{B}{x+1} + \frac{C}{(x+1)^{2} }dx\\\\\\ Observando ~~a~~ tabela ~~de ~~derivadas~~encontramos.\\\\\\\ \displaystyle\int \frac{5 x^{2} +20x+6}{ x^{3} 3+2x+x}=6ln|x|-ln|x+1|- \frac{9}{(x+1)} +c$

Bom dia! 

Bons estudos!