Question

me ajudem a resolver essa integral
∫x²∛3x³+7 dx

Answer 1

Olá


$\displaystyle \mathsf{\int x^2 \sqrt[3]{3x^2+7} }~dx$


Podemos resolver essa integral usando uma substituição 'udu' simples

$\displaystyle \mathsf{u=3x^3+7}\\\mathsf{du=9x^2dx \qquad\qquad \Longrightarrow \qquad \qquad \frac{1}{9}du=x^2dx }$


Substituindo na integral

$\displaystyle \mathsf{\frac{1}{9}\int \sqrt{u} ~du}\\\\\\\mathsf{\frac{1}{9}\int u^ \frac{1}{2}du }\\\\\\\mathsf{= \frac{1}{9}\cdot \frac{u^{ \frac{1}{2}+1 }}{ \frac{1}{2}+1 } }\\\\\\\mathsf{=\frac{1}{9}\cdot \frac{u^{\frac{3}{2} }}{ \frac{3}{2} } }\\\\\\\mathsf{=\frac{1}{9}\cdot \frac{2u^{ \frac{3}{2} }}{3} +C}}}$


Voltando com a variavel 'x' no lugar do 'u'


$\displaystyle \mathsf{u=3x^3+7}\\\\\\\mathsf{ = \frac{2(3x^3+7)^ \frac{3}{2} }{27} +C}\\\\\\\text{Voltando para forma de raiz}\\\\\\\mathsf{\boxed{ = \frac{2 \sqrt{(3x^3+7)^3} }{27} +C}}$