Question

Calculo
..............

∫(x²+1)^4

Answer 1

Olá


$\displaystyle \mathsf{\int (x^2+1)^4dx}$


Não há nenhum método de integração que possamos utilizar, então só nos resta expandir esse polinômio

$\displaystyle\mathsf{(x^2+1)^4}\\\\\mathsf{(x^2+1)^2\cdot(x^2+1)^2}\\\\\mathsf{(x^4+2x^2+1)\cdot(x^4+2x^2+1)}\\\\\mathsf{x^8+2x^6+x^4+2x^6+4x^4+2x^2+x^4+2x^2+1}\\\\\boxed{\mathsf{x^8+4x^6+6x^4+4x^2+1}}~~~~~ ~\longleftarrow\text{Vamos integrar esse polinomio}\\\\\\\mathsf{\int (x^8+4x^6+6x^4+4x^2+1)dx}\\\\\\=\mathsf{ \frac{x^{8+1}}{8+1} + \frac{4x^{6+1}}{6+1} + \frac{6x^{4+1}}{4+1} + \frac{4x^{2+1}}{2+1} +x}$

$\displaystyle=\boxed{\boxed{\mathsf{ \frac{x^{9}}{9} + \frac{4x^{7}}{7} + \frac{6x^{5}}{5} + \frac{4x^{3}}{3} +x+C}}}$