Question

integral(5x-3)^3dx solucao com integral

Answer 1

$\displaystyle\int\!(5x-3)^3\,dx\\\\\\ =\int\!\frac{1}{5}\cdot 5\cdot (5x-3)^3\,dx\\\\\\ =\frac{1}{5}\int\!(5x-3)^3\cdot 5\,dx~~~~~~\mathbf{(i)}$


Faça a seguinte substituição:

$5x-3=u~~\Rightarrow~~5\,dx=du$


Substituindo, a integral $\mathbf{(i)}$ fica

$\displaystyle=\frac{1}{5}\int\!u^3\,du\\\\\\ =\frac{1}{5}\cdot \dfrac{u^{3+1}}{3+1}+C\\\\\\ =\frac{1}{5}\cdot \dfrac{u^4}{4}+C\\\\\\ =\frac{u^4}{20}+C\\\\\\ =\frac{(5x-3)^4}{20}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int\!(5x-3)^3\,dx=\frac{1}{20}(5x-3)^4+C \end{array}}$


Bons estudos! :-)

Answer 2

Resposta:

∫ (5x-3)³ dx

por substituição ==>u=5x-3 ==>du=5dx

∫ (u)³ du/5  =[u⁴/4] *(1/5) + constante=u⁴/20 + constante

Como u=5x-3

∫ (5x-3)³ dx  =(5x-3)⁴/20 + constante