Guess $an$ antiderivative for the integrand function. Validate your guess $by$ differentiation and then evaluate the given definite integral. $(Hint$: Keep $in$ mind the Chain Rule $in$ guessing $an$ antiderivative.$)$

${\displaystyle \underset{5}{\overset{7}{}}}\frac{xdx}{\sqrt[\phantom{2}]{1+x^2}}$