Question

Find the value of \( \int x^2 e^{x^3} dx \).

Hint:

Use substitution to simplify integrals, setting \( u \) as the exponent term when possible.

Answer 1

Let \( I = \int x^2 e^{x^3} dx \). Using substitution, let: \[ u = x^3 \Rightarrow du = 3x^2 dx. \] Rewriting, \[ \frac{du}{3} = x^2 dx. \] Thus, the integral becomes: \[ I = \int e^u \frac{du}{3} = \frac{1}{3} \int e^u du. \] Since \( \int e^u du = e^u \), \[ I = \frac{1}{3} e^u + C. \] Substituting back \( u = x^3 \), \[ I = \frac{1}{3} e^{x^3} + C. \]