Question

Usando o método de substituição determine a integral 0∫¹ (x³+2)elevado a 4. x²dx|?

Answer 1

$\\ \int_{0}^{1}(x^3 + 2)^4 \cdot x^2 \, dx = \\\\ \begin{cases} x^3 + 2 = \lambda \\ d\lambda = 3x^2 \, dx \Rightarrow \boxed{x^2 \, dx = \frac{d\lambda }{3}}\end{cases} \\\\\\ \int_{0}^{1} \lambda^4 \cdot \, \frac{d\lambda }{3} = \\\\\\ \frac{1}{3} \int_{0}^{1} \lambda^4 \, d\lambda = \left [ \frac{1}{3} \cdot \frac{\lambda ^5}{5} \right ]_{0}^{1} = \left [ \frac{\lambda ^5}{15} \right ]_{0}^{1} =$

 

$\\ \left [ \frac{(x^3 + 2)^5}{15} \right ]_{0}^{1} = \\\\\\ \frac{(1 + 2)^5}{15} - \frac{(0 + 2)^5}{15} = \\\\ \frac{243 - 32}{15} = \\\\ \boxed{\boxed{\frac{211}{15}}}$