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integral de (2x^2+x)/((4x^3+3x^2)^2)

Solução

\int \frac{2 x ^{2} + x}{( 4 x ^{3} + 3 x ^{2} ) ^{2}} d x

Solução

\frac{2}{27 x} - \frac{1}{18 x ^{2}} - \frac{8}{27 ( 4 x + 3 )} + C

Passos da solução

\int \frac{2 x ^{2} + x}{( 4 x ^{3} + 3 x ^{2} ) ^{2}} d x

Tirar a fração parcial de

\frac{2 x ^{2} + x}{( 4 x ^{3} + 3 x ^{2} ) ^{2}} : - \frac{2}{27 x ^{2}} + \frac{1}{9 x ^{3}} + \frac{32}{27 ( 4 x + 3 ) ^{2}}

= \int - \frac{2}{27 x ^{2}} + \frac{1}{9 x ^{3}} + \frac{32}{27 ( 4 x + 3 ) ^{2}} d x

Aplicar a regra da soma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{2}{27 x ^{2}} d x + \int \frac{1}{9 x ^{3}} d x + \int \frac{32}{27 ( 4 x + 3 ) ^{2}} d x

\int \frac{2}{27 x ^{2}} d x = - \frac{2}{27 x}

\int \frac{1}{9 x ^{3}} d x = - \frac{1}{18 x ^{2}}

\int \frac{32}{27 ( 4 x + 3 ) ^{2}} d x = - \frac{8}{27 ( 4 x + 3 )}

= - (- \frac{2}{27 x}) - \frac{1}{18 x ^{2}} - \frac{8}{27 ( 4 x + 3 )}

Simplificar

= \frac{2}{27 x} - \frac{1}{18 x ^{2}} - \frac{8}{27 ( 4 x + 3 )}

Adicionar uma constante à solução

= \frac{2}{27 x} - \frac{1}{18 x ^{2}} - \frac{8}{27 ( 4 x + 3 )} + C

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