Question

Alguém poderia me ajudar a resolver essa Integral Indefinida:

$\frac{(x-2)^{\frac{1}{2} } }{(x + 1)} dx$

Answer 1

Importante saber que: $\int\frac{dx}{x^2+a^2}=\frac{1}{a}.\arctan(\frac{x}{a})+C$

$\int \frac{(x-2)^\frac{1}{2}}{x+1}dx\\\\\int \frac{\sqrt{x-2}}{x+1}dx\\\\\\u=\sqrt{x-2}\,\,\Rightarrow\,\,du=\frac{dx}{2\sqrt{x-2}}\\\\u^2=x-2\,\,\Rightarrow\,\,x=u^2+2\\\\\\\int\frac{\sqrt{x-2}.2.\sqrt{x-2}du}{x+1}\\\\\int\frac{2(x-2)}{x+1}du\\\\\int\frac{2(u^2+2-2)}{u^2+2+1}du\\\\2\int\frac{u^2}{u^2+3}$

Realizando a divisão entre os polinômios:

$2\int1-\frac{3}{u^2+3}du\\\\2u-6\int\frac{du}{u^2+3}\\\\2u-6.\frac{1}{\sqrt{3}}.\arctan(\frac{u}{\sqrt{3}})+C\\\\2\sqrt{x-2}-2\sqrt{3}.\arctan(\frac{\sqrt{x-2}}{\sqrt{3}})+C$