Question

Resolva a Integral Indefinida:

b) integral de x^3 * raiz de (x^2 + 1) dx

Answer 1

Resolver a integral indefinida:

$\displaystyle\int\!x^3\sqrt{x^2+1}\,dx\\\\\\ =\int\!x\cdot x^2\sqrt{x^2+1}\,dx\\\\\\ =\int\!\frac{1}{2}\cdot 2x\cdot x^2\sqrt{x^2+1}\,dx\\\\\\ =\frac{1}{2}\int\!x^2\sqrt{x^2+1}\cdot 2x\,dx\qquad\mathbf{(i)}$


Faça a seguinte substituição:

$\sqrt{x^2+1}=u\\\\ x^2+1=u^2~~\Rightarrow~~\left\{\! \begin{array}{rcl} x^2&\!\!\!=\!\!\!&u^2-1\\ 2x\,dx&\!\!\!=\!\!\!&2u\,du \end{array} \right.$


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

$=\displaystyle\frac{1}{\diagup\!\!\!\! 2}\int\!(u^2-1)\cdot u\cdot \diagup\!\!\!\! 2u\,du\\\\\\ =\int\!(u^2-1)\cdot u^2\,du\\\\\\ =\int\!(u^4-u^2)\,du\\\\\\ =\frac{u^{4+1}}{4+1}-\frac{u^{2+1}}{2+1}+C$

$=\dfrac{u^5}{5}-\dfrac{u^3}{3}+C\\\\\\ =\dfrac{(\sqrt{x^2+1})^5}{5}-\dfrac{(\sqrt{x^2+1})^3}{3}+C\\\\\\ =\dfrac{1}{5}\,(x^2+1)^{5/2}-\dfrac{1}{3}\,(x^2+1)^{3/2}+C\qquad\checkmark$


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Bons estudos! :-)


Tags: integral indefinida irracional raiz quadrada substituição mudança variável cálculo integral