Question

Evaluate the integral \[ \int x^5 e^{x^3} \,dx. \]

• A. \( \frac{e^{x^3}}{3} (x^3 - 1) + C \)
• B. \( \frac{e^{x^3}}{5} (x^5 - 1) + C \)
• C. \( \frac{e^{x^3}}{4} (x^4 - 1) + C \)
• D. \( \frac{e^{x^3}}{3} (x^5 - 1) + C \)
• E. \( \frac{x^3 e^{x^3}}{3} + C \)

Hint:

For integrals involving exponentials and polynomials, try substitution and integration by parts.

Answer 1

The Correct Option is A

Step 1: Substitution Let \( u = x^3 \), so that: \[ \frac{du}{dx} = 3x^2 \Rightarrow du = 3x^2 dx. \] Rewriting the integral: \[ \int x^5 e^{x^3} dx = \int x^3 x^2 e^{x^3} dx. \] Since \( x^3 = u \), and \( x^2 dx = \frac{du}{3} \): \[ \int u e^u \frac{du}{3}. \] Step 2: Integration by Parts Using integration by parts, where: \[ \int u e^u du = (u - 1)e^u + C. \] Applying this: \[ \frac{1}{3} (x^3 - 1) e^{x^3} + C. \] Thus, the correct answer is: \[ \frac{e^{x^3}}{3} (x^3 - 1) + C. \]