Question

Resolva a Integral abaixo???

Answer 1

Olá


Integração por substituição


$\displaystyle \mathsf{ \int\limits^1_0 {(12x^2+8x+3)\cdot (4x^3+4x^2+3x+5)^3} \, dx }\\\\\\\\\text{Fazendo a substituicao 'udu'}\\\\\mathsf{u=4x^3+4x^2+3x+5}\\\\\mathsf{du=12x^2+8x+3}\\\\\\\text{Substituindo na integral}\\\\\\ \mathsf{ \int\limits^1_0 {u^3du} }\\\\\\\mathsf{\left( \frac{u^{3+1}}{3+1} \right)\bigg|^1_0}\\\\\\\mathsf{\left( \frac{u^4}{4} \right)\bigg|^1_0}\\\\\\\mathsf{Lembrando~ que: \qquad u=4x^3+4x^2+3x+5}$

$\displaystyle \mathsf{ \left(\frac{(4x^3+4x^2+3x+5)^4}{4}\right)\bigg|^1_0 }\\\\\\\\\mathsf{\left(\frac{(4(1)^3+4(1)^2+3(1)+5)^4}{4}\right)~-~\left(\frac{(4\cdot(0)^3+4\cdot(0)^2+3\cdot(0)+5)^4}{4}\right)}\\\\\\\\\mathsf{ \frac{16^4}{4}~-~ \frac{5^4}{4} }\\\\\\\mathsf{ \frac{65536}{4}~-~ \frac{625}{4} }\\\\\\\mathsf{ \frac{65536-625}{4} }\\\\\\\\\boxed{\mathsf{ \frac{64911}{4} }}\qquad\qquad\qquad\Longleftarrow \qquad\qquad\text{Resposta}$