Question

alguém resolve a integral?
$\int\limits { \frac{1}{(1+x) \sqrt{x} } } \, dx$

Answer 1

Calcular a integral indefinida:

     $\displaystyle\int\frac{1}{(1+x)\sqrt{x}}\,dx$


Faça a seguinte substituição:

     $\sqrt{x}=u\quad\Rightarrow\quad\left\{\! \begin{array}{l} x=u^2\\\\ dx=2u\,du \end{array} \right.$


Substituindo, a integral fica

     $\displaystyle\int\frac{1}{(1+u^2)u}\cdot 2u\,du\\\\\\ =2\int\frac{1}{1+u^2}\,du\\\\\\ =2\,\mathrm{arctg}(u)+C$

     $=2\,\mathrm{arctg}(\sqrt{x})+C\quad\longleftarrow\quad\textsf{resposta.}$


Bons estudos! :-)