Question

qual a integral indefinida da função ∫dx/√x³ dx ?
a) - 1/3x³+c
b)-2/√x+c
c) -2x/√x+c
d) - 1/5x⁵+c
qual a primitiva que resulta na integral ∫x³×√xdx ?
a)9/2√x⁹+c
b)9 ⁹√x²/2+c
c)2 ⁹√x²/9+c
d)2/9 √x⁹+c​

Answer 1

Resposta:

a) ∫dx/√x³

√x³ =x^(-3/2)

∫x^(-3/2)   dx

=x^(-3/2+1) / (-3/2+1)   +c

=x^(-1/2)/(-1/2) + c

=-2*x^(-1/2) + c

=-2/√x  +c

Letra B

b)

∫x³*√x  dx

∫x³* x^(1/2) dx

∫x^(3+1/2)  dx

∫x^(7/2)  dx

= x^(7/2+1) /(7/2+1)  +c

=x^(9/2) / (9/2) + c

=2*x^(9/2) /9  + c

=(2/9) * √x⁹ + c

Letra D

Answer 2

                                           $\boxed{\int\limits {\frac{dx}{\sqrt{x^3} } =?} }$

=                                          $\int\limits{\frac{1}{x^{\frac{3}{2} }} } \, dx=\\\\\\\int\limits {x^{-\frac{3}{2} }} \, dx = \\\\\\\frac{x^{-\frac{3}{2} +1}}{-\frac{3}{2}+1 } +C =\\\\\\\frac{x^{\frac{-1}{2} }}{\frac{-1}{2} } +C =\\\\\\2\times \frac{x^{\frac{-1}{2} }}{-1} +C=\\\\\\ -2.\frac{1}{x^{\frac{1}{2} }} +C =\\\\\\\frac{-2}{\sqrt{x} } +C$

Resposta: b)

                                              $\boxed{\int\limits {x^3.\sqrt{x} } \, dx }$

                                 

                                                $\int\limits {x^3.\sqrt{x} } \, dx =\\\\\\ \int\limits {x^3.x^{\frac{1}{2} }} \, dx =\\\\\\ \int\limits {x^{3+\frac{1}{2} }} \, dx =\\\\ \int\limits {x^{\frac{7}{2} }} \, dx =\\\\\frac{x^{\frac{7}{2} +1}}{\frac{7}{2} +1} +C=\\\\\frac{x^{\frac{9}{2} }}{\frac{9}{2} } +C = \\\\ \frac{2}{9} .\sqrt{x^9} +C$

Resposta: d)