Question

Calcule a integral por substituição de variável

Answer 1

Olá!
 
     Chame $4+3x$   de  $u$  . Daí, 

$4+3x=u\Rightarrow \dfrac{d(u)}{dx}=\dfrac{d(4+3x)}{dx}\Rightarrow \dfrac{du}{dx}=3\Rightarrow dx=\dfrac{1}{3}du \\ \\ \therefore \\ \\ \displaystyle{\int}\, \dfrac{dx}{4+3x} =\displaystyle{\int} \dfrac{1}{u}\cdot \dfrac{1}{3}du=\dfrac{1}{3}\cdot \displaystyle{\int} \dfrac{1}{u}du = \dfrac{1}{3}\ln|u|+k = \\ \\ = \dfrac{1}{3}\ln|4+3x|+k,\,k\in\mathbb{R}$

Bons estudos!

Answer 2

$\sf \displaystyle \int \frac{dx}{4+3x}\\\\\\=\int \frac{1}{3u}du\\\\\\=\frac{1}{3}\cdot \int \frac{1}{u}du\\\\\\=\frac{1}{3}ln \left|u\right|\\\\\\=\frac{1}{3}ln \left|4+3x\right|\\\\\\\to \boxed{\sf =\frac{1}{3}ln \left|4+3x\right|+C}$