Please answer the question listed below and show work! Thank you **Problem Statement:**Evaluate the definite integral:\[\int_{0}^{3} x^3 \, e^{\frac{x}{3}} \, dx\]**Solution Outline:**To solve this integral, we might consider using integration techniques such as integration by parts. Integration by parts is given by the formula:\[\int u \, dv = uv - \int v \, du\]Choose \( u = x^3 \) and \( dv = e^{\frac{x}{3}} \, dx \).### Steps:1. **Differentiate** \( u = x^3 \): \[ du = 3x^2 \, dx \]2. **Integrate** \( dv = e^{\frac{x}{3}} \, dx \): \[ v = 3e^{\frac{x}{3}} \]3. Substitute back into the integration by parts formula: \[ \int x^3 \, e^{\frac{x}{3}} \, dx = x^3 \cdot 3e^{\frac{x}{3}} - \int 3 \cdot 3x^2 \cdot e^{\frac{x}{3}} \, dx \]4. Simplify and compute the remaining integral: \[ = 3x^3 \cdot e^{\frac{x}{3}} - 9 \int x^2 \cdot e^{\frac{x}{3}} \, dx \]5. **Repeat** the integration by parts for the term \(\int x^2 \cdot e^{\frac{x}{3}} \, dx\).6. **Evaluate** the definite integral from \(0\) to \(3\).This method will require repeated use of integration by parts until the integral becomes manageable and solvable. Make sure to check boundaries and substitute them back into the integrated result to obtain the final answer.

Name: Please answer the question listed below and show work! Thank you **Pro
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: Please answer the question listed below and show work! Thank you **Problem Statement:**Evaluate the definite integral:\[\int_{0}^{3} x^3 \, e^{\frac{x}{3}} \, dx\]**Solution Outline:**To solve this integral, we might consider using integration techni