Question

Faça a integração pelo método da substituição

∫$\frac{x}{(x^2+5)^3} dx$

Answer 1

$\int\frac{x}{(x^2+5)^3}dx \\\\ g=x^2+5\implies dg=2xdx\implies dx=\frac{dg}{2x}\\\\ \int \frac{1}{2}g^{-3}dg=\frac{1}{2}\int g^{-3}dg=\frac{1}{2}-2g^{-2}+c\implies -\frac{1}{g^2}+c\\\\\int\frac{x}{(x^2+5)^3}dx=-\frac{1}{(x^2+5)^2}+c$

Answer 2

Olá

$\displaystyle \int \frac{x}{(x^2+5)^3} dx \\ \\ \\ \mathsf{u=x^2+5} \\ \mathsf{du=2xdx} \\\\ \mathsf{ \frac{1}{2x}du=dx }~~~~ ~~~~ ~~\longleftarrow\text{Passei o 2x para o outro lado dividindo} \\ \\ \\ \text{substituindo na integral} \\ \\ \\ \int \frac{\diagup\!\!\!\!x}{u^3} \cdot \frac{1}{2\diagup\!\!\!\!x} du \\ \\ \\ \int \frac{du}{2u^3}$

$\displaystyle \frac{1}{2} \int u^{-3}du~~~~~ ~~~ ~~~\longleftarrow \text{Tirei a constante a integral e passei o 'u'} \\ ~ ~~~~~~ ~~~~~~ ~~ ~~~~~~~~ ~~ ~~~~~ ~~~\text{para o numerador} \\ \\ \\ = \frac{1}{2}\cdot \left.\left(\dfrac{u^{-3+1}}{-3+1}\right)$

$\displaystyle = \frac{1}{2} \cdot \frac{u^{-2}}{-2} \\ \\ \\ = -\frac{1}{4u^{2}} \\ \\ \\ \text{voltando com }x^2+5\text{ no lugar do 'u'}\\\\\\=\boxed {-\frac{1}{4(x^2+5)^2} +C}$

$\displaystyle \\ \\ \\ \text{Outra opcao e deixar o 'u' no numerador, dai a resposta fica} \\ \\ \\ = \boxed{-\frac{(x^2+5)^{-2}}{4} +C} \\ \\ \\ \text{Ambas notacoes estao corretas.}$