Question

Resolva a integral definida abaixo utilizando a técnica da “u”-substituição e marque a alternativa correta:
f0² [ 2x (x² + 1)³ ] dx

(a)352
(b)258
(c)562
(d)426
(e)156​

Answer 1

$\displaystyle \sf \int\limits^2_0 (x^2+1)^3\ 2x\ dx \\\\\\ \text{Fa{\c c}amos} : \\\\ x^2+1 = u \to 2x\ dx = du \\\\ Da{\'i} } :\\\\ \int\limits^2_0 u^3 dx = \left \frac{u^4}{4} \right|\limits^2_0 \\\\\\ \left \frac{(x^2+1 )^4}{4} \right|\limits^2_0 = \frac{(2^2+1)^4}{4}-\frac{(0^2+1)^4}{4} \\\\\\ \frac{5^4}{4}-\frac{1}{4} = \frac{625-1}{4} = \frac{624}{4} = 156 \\\\\\ Portanto : \\\\ \boxed{\sf \int\limits^2_0 (x^2+1)^3\ 2x\ dx = 156 \ }\checkmark$

letra e