Calcule \( \int_0^1 x (1 - x^2)^{\frac{3}{2}} \, dx \). A) \( \frac{1}{4} \) B) \( \frac{1}{5} \) C) \( \frac{1}{6} \) D) \( \frac{1}{8} \)

Para resolver a integral \( \int_0^1 x (1 - x^2)^{\frac{3}{2}} \, dx \), podemos usar a substituição \( u = 1 - x^2 \). Assim, temos: 1. Derivando, obtemos \( du = -2x \, dx \) ou \( dx = -\frac{du}{2x} \). 2. Quando \( x = 0 \), \( u = 1 \) e quando \( x = 1 \), \( u = 0 \). Substituindo na integral, temos: \[ \int_0^1 x (1 - x^2)^{\frac{3}{2}} \, dx = \int_1^0 x u^{\frac{3}{2}} \left(-\frac{du}{2x}\right) = \frac{1}{2} \int_0^1 u^{\frac{3}{2}} \, du \] Agora, calculamos a integral \( \int_0^1 u^{\frac{3}{2}} \, du \): \[ \int u^{\frac{3}{2}} \, du = \frac{u^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} u^{\frac{5}{2}} \bigg|_0^1 = \frac{2}{5} \] Portanto, substituindo de volta: \[ \frac{1}{2} \cdot \frac{2}{5} = \frac{1}{5} \] Assim, a resposta correta é: B) \( \frac{1}{5} \)