Question

Calcule a integral ∫(2x+3)/√(4-x²) dx

Answer 1

$Ol\acute{a}~~\mathbb{OZENI} \\ \\ Seja: \\ \int( \frac{2x+3}{ \sqrt{4- x^{2} } })dx \\ \\ Resolu\zeta \tilde{a}o \\ \\ Temos~que:~\boxed{2cos \alpha = \sqrt{4- x^{2} } }~, ~\boxed{x=2sen \alpha }~~e~~\boxed{dx=2cos \alpha d \alpha} \\ Substituindo~temos: \\ \\ \int\bigg( \frac{2(2sen \alpha )+3}{2cos \alpha } \bigg)2cos \alpha d \alpha ~~--\ \textgreater \ simplificando~fica: \\ \\ \int(4sen \alpha +3)d \alpha$

$\int4sen \alpha d \alpha +\int 3d \alpha \\ \\ -4cos \alpha +3 \alpha +C ~~~--\ \textgreater \ Se~sabe~que:\begin{cases} cos \alpha = \frac{ \sqrt{4- x^{2} } }{2} \\ \\ \alpha = cos^{-1} \frac{ \sqrt{4- x^{2} } }{2} \end{cases} \\ \\ Substituindo~temos: \\ \\ -\not 4 \bigg(\frac{ \sqrt{4- x^{2} } }{\not2} \bigg)+3\bigg( cos^{-1} \frac{ \sqrt{4- x^{2} } }{2} \bigg)+C \\ \\ \\\boxed{\boxed{-2 \sqrt{4- x^{2} } +3sec \frac{ \sqrt{4- x^{2} } }{2}+C }}$

$\mathbb{iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Espero~ter~ajudado!!$