Question

integrando por substituição ∫x.√x²+1dx, obtém-se :
escolha uma:
a) √(x²+1)³
b) √(x²+1)³/3
c)√2x
d) √x²+1 /3

Answer 1

∫$x. \sqrt{x^2+1} dx$

$u = x^2 + 1\\\0u'=2xdx$

∫$\frac{1}{2}.u^{ \frac{1}{2}}du$

$\dfrac{1}{2}. (\dfrac{ u^{ \frac{1}{2}+1 } }{ \frac{1}{2}+1 } )$

$\dfrac{1}{2}. \dfrac{ u^{ \frac{3}{2} } }{ \frac{3}{2} } \\\\ \dfrac{ u^{ \frac{3}{2} } }{3} \\\\ \dfrac{ \sqrt{u^3} }{3}\\\\\ \dfrac{ \sqrt{(x^2+1)^3} }{3} +C$