Question

e) \\( \int x^2 e^{-x^3} dx = \\)

Explanation:

Step1: Choose Substitution

Let \( u = -x^3 \). Then, find the derivative of \( u \) with respect to \( x \): \( \frac{du}{dx} = -3x^2 \), so \( dx = \frac{du}{-3x^2} \) (when \( x^2
eq0 \)).

Step2: Substitute into Integral

Substitute \( u = -x^3 \) and \( dx = \frac{du}{-3x^2} \) into the integral \( \int x^2 e^{-x^3} dx \):

$$
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$$

The \( x^2 \) terms cancel out (for \( x
eq0 \), and we can check the case \( x = 0 \) separately, but the antiderivative will hold for all \( x \) in the domain of the original function).

Step3: Integrate \( e^u \)

We know that the integral of \( e^u \) with respect to \( u \) is \( e^u + C \) (where \( C \) is the constant of integration). So,

$$ -\frac{1}{3}\int e^{u} du = -\frac{1}{3}e^{u} + C $$

Step4: Substitute Back \( u \)

Substitute back \( u = -x^3 \) into the expression:

$$ -\frac{1}{3}e^{-x^3} + C $$

Answer:

\( -\frac{1}{3}e^{-x^3} + C \) (where \( C \) is the constant of integration)