Question

Resolva a seguinte integral
∫5x√4-3x² dx

Answer 1

Pretendemos calcular o integral:

$\displaystyle\int 5x \sqrt{4 - 3x^2}\textrm{ d}x.$

Começamos por fazer a mudança de variável:

$u = 4 - 3x^2 \implies \textrm{d}u = -6x \textrm{ d}x \implies x \textrm{ d}x = -\dfrac{\textrm{d}u}{6}.$

Assim, vem:

$\displaystyle\int 5x \sqrt{4 - 3x^2}\textrm{ d}x = 5\int\underbrace{\sqrt{4 - 3x^2}}_{=\sqrt{u}}\underbrace{x\textrm{ d}x}_{=-\frac{\textrm{d}u}{6}} = -\dfrac{5}{6}\int\sqrt{u}\textrm{ d}u.$

Como $\sqrt{u} = u^{1/2}$, o integral é simples:

$\displaystyle\int u^{1/2}\textrm{ d}u = \dfrac{u^{3/2}}{3/2} + C= \dfrac{2}{3}u^{3/2} + C, \quad \textrm{com } C \in \mathbb{R}.$

Portanto:

$\displaystyle-\dfrac{5}{6}\int\sqrt{u}\textrm{ d}u = -\dfrac{5}{6}\times\left(\dfrac{2}{3}u^{3/2} + C\right) = -\dfrac{10}{18}u^{3/2} - \dfrac{5}{6}C = -\dfrac{5}{9}u^{3/2} + K,$

com $K = -\frac{5}{6}C$.

Desfazendo a mudança de variável, obtemos o resultado final:

$\boxed{\displaystyle\int 5x \sqrt{4 - 3x^2}\textrm{ d}x = -\dfrac{5}{9}(4 - 3x^2)^{3/2} + K, \quad \textrm{com } K \in \mathbb{R}}.$