Question

Calcule a integral por substituição de variável

obs : O denominador é 5 - x

Answer 1

$I=\displaystyle\int\!\dfrac{dx}{5-x}\\\\\\ =-\int\!\dfrac{-1}{5-x}\,dx~~~~~~\mathbf{(i)}$


Faça a seguinte substituição:

$5-x=u~~\Rightarrow~~-1\,dx=du$


Substituindo, a integral $\mathbf{(i)}$ fica

$=\displaystyle-\int\!\dfrac{du}{u}\\\\\\ =-\,\mathrm{\ell n}\!\left|u\right|+C\\\\ =-\,\mathrm{\ell n}\!\left|5-x\right|+C\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int\!\dfrac{dx}{5-x}=-\,\mathrm{\ell n}\!\left|5-x\right|+C \end{array}}$


Bons estudos! :-)

Answer 2

$\sf \displaystyle \int \frac{dx}{5-x}dx\\\\\\=\int \:-\frac{1}{u}du\\\\\\=-\int \frac{1}{u}du\\\\\\=-ln \left|u\right|\\\\\\=-ln \left|5-x\right|\\\\\\\to \boxed{\sf =-ln \left|5-x\right|+C}$