Question

Calcule a integral por substituição de variável

Answer 1

$\displaystyle I=\int\!e^{2x}\,dx\\\\\\ =\int\!\frac{1}{2}\cdot 2e^{2x}\,dx\\\\\\ =\frac{1}{2}\int\!e^{2x}\cdot 2\,dx~~~~~~\mathbf{(i)}$


Faça a seguinte substituição:

$2x=u~~\Rightarrow~~2\,dx=du$


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

$\displaystyle=\frac{1}{2}\int\!e^u\,du\\\\\\ =\frac{1}{2}\,e^u+C\\\\\\ =\frac{1}{2}\,e^{2x}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int\!e^{2x}\,dx=\frac{1}{2}\,e^{2x}+C \end{array}}$


Bons estudos! :-)

Answer 2

$\sf \displaystyle \int \:e^{2x}dx\\\\\\=\int \:e^u\frac{1}{2}du\\\\\\=\frac{1}{2}\cdot \int \:e^udu\\\\\\=\frac{1}{2}e^u\\\\\\=\frac{1}{2}e^{2x}\\\\\\\to \boxed{\sf =\frac{1}{2}e^{2x}+C}$