Question

Resolva as integrais indefinidas:

l)∫xdx/√1-x²

Answer 1

$\int \frac{x}{\sqrt{1-x^2}} \; dx$

Substituição
$\boxed{\boxed{u = 1-x^2}}\\\\ \frac{du}{dx}= -2x\\\\\boxed{\boxed{dx = \frac{du}{-2x}}}$

substituindo na integral
$\int \frac{x}{\sqrt{u} }* \frac{du}{-2x} \\\\ = - \frac{1}{2} \int \frac{\not x}{\sqrt{u} }* \frac{du}{-2 \not x} \\\\ = - \frac{1}{2} \int \frac{1}{\sqrt{u}} \;du \\\\\ = - \frac{1}{2} \int u^{- \frac{1}{2} } \;du =\frac{-1}{2} \left [ \frac{u^{\frac{-1}{2}+1 }}{\frac{-1}{2} +1} +C \right] = \frac{-1}{2}\left[ \frac{u^{ \frac{1}{2} }}{ \frac{1}{2} } +C\right] = \frac{-1}{2}\left[ 2\sqrt{u}+C\right] \\\\ \boxed{\boxed{=-\sqrt{1-x^2}+C}}$