Respostas

2014-06-22T18:45:30-03:00
 \int\limits {\frac{1}{x^3+2}* x^2 } \, dx = \boxed{\int\limits {\frac{x^2}{x^3+2} } \, dx }

temos:
\boxed{u=x^3+2}\\\\du=3x^2.dx\\\\\ \boxed{\frac{du}{3x^2} =dx}

a integral fica
 \int\limits { \frac{x^2}{u} } \, dx

substituindo o valor de  (dx)

 \int\limits { \frac{x^2}{u} } \, . \frac{du}{3x^2} =  \int\limits { \frac{1}{u} } \,. \frac{du}{3}\\\\ = \frac{1}{3} \int\limits { \frac{1}{u} } \, .du  \\\\\\\\\boxed{ \frac{1}{3}  \int\limits u^{-1}.du}

a integral de u^{-1}=ln|u|

então ficamos com
 \frac{1}{3} *ln|u|= \frac{ln|u|}{3}

substituindo o valor de u
\frac{ln|x^3+2|}{3}

resposta:
\boxed{\boxed{ \int\limits { \frac{x^2}{x^3+2} } \, dx =\frac{ln|x^3+2|}{3}+C}}

C = constante
2 5 2