Question

Calcule a seguinte integral indefinida

Answer 1

$\displaystyle\mathsf{\int\sqrt{1-x^4}x^3\,dx=-\dfrac{1}{4}\int\sqrt{1-x^4}-4x^3\,dx}$

faça

$\mathsf{u=1-x^4\implies~du=-4x^3~dx}$

$\displaystyle\mathsf{-\dfrac{1}{4}\int\sqrt{1-x^4}-4x^3\,dx=-\dfrac{1}{4}\int\sqrt{u}du=-\dfrac{1}{4}\cdot\dfrac{2}{3}u^{\frac{3}{2}}+k}\\\displaystyle\mathsf{\int\sqrt{1-x^4}x^3\,dx=-\dfrac{1}{6}(1-x^4)^{\frac{3}{2}}+k}$