Question

$Resolva:\int\limits^1_0 {x^2 e^-x^3} \, dx$

Answer 1

$\displaystyle \mathsf{ \int_0^1 x^2.e^{-x^3}dx}$

Calcularemos primeiro a integral indefinida. Vamos chamar -x³ de u:

$\displaystyle \mathsf{ \int_0^1 x^2.e^{-x^3}dx}$

$\displaystyle \mathsf{-3x^{2} dx = du} \\ \displaystyle \mathsf{ {x}^{2}dx = - \frac{du}{3} }$

Substituindo no integrando:

$\displaystyle \mathsf{ \int - \frac{1}{3} e^{u}du }$

Calculando a integral:

$\displaystyle \mathsf{ - \frac{1}{3} \int e^{u}du = - \frac{1}{3}e^{u} + c }$

Como u = -x³:

$\displaystyle \mathsf{ - \frac{1}{3} \int e^{u}du = - \frac{1}{3}e^{ - {x}^{3} } + c }$

Agora vamos calcular a definida.

$\displaystyle \mathsf{ - \frac{1}{3}e ^{ - 1^{3} } + c - ( - \frac{1}{3}e^{ 0^{3}} + c) = - \frac{1}{3e} + \frac{1}{3} }$