Question

Qual a integral de 1/(x+1)⁵ dx​

Answer 1

$\displaystyle \int \frac{1}{(\text x+1)^5}\text{dx}$

Façamos :

$\text x+1=\text u$

$\text{dx}=\text{du}$

Substituindo na integral :

$\displaystyle \int \frac{1}{\text u^5}\text{du} \\\\\\ \int \text u^{-5}\text{du} \\\\\\\frac{\text u^{-5+1}}{-5+1}+\text C \\\\\\ \frac{\text u^{-4}}{-4}+\text C \\\\\\ \frac{-1}{4.\text u^{-4}}+\text C$

Desfazendo a troca de variável :

$\displaystyle \frac{-1}{4.(\text x+1)^{4}} + \text C$

Portanto :

$\huge\boxed{\displaystyle \int \frac{1}{(\text x+1)^5}\text{dx} =\displaystyle \frac{-1}{4.(\text x+1)^{4}} + \text C\ }\checkmark$