Questions: Evaluate the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) ∫ from 0 to 4 of x e^(x / 3) dx

Evaluate the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.)

∫ from 0 to 4 of x e^(x / 3) dx

Transcript text: Evaluate the definite integral. Use a graphing utility to verify your result. (Round your answer to three decimal places.) \[ \int_{0}^{4} x e^{x / 3} d x \]

Solution

Solution Steps

Step 1: Identify the Type of Function

The function $f(x) = x e^{\frac{x}{3}}$ is identified for integration.

Step 2: Simplification

The function is simplified (if possible) to $f(x) = x e^{\frac{x}{3}}$ for easier integration.

Step 3: Integration

The antiderivative of $f(x)$ is $F(x) = \left(3 x - 9\right) e^{\frac{x}{3}}$.

Step 4: Evaluate the Definite Integral

The definite integral of $f(x)$ from $0$ to $4$ is evaluated as $20.381$.

Final Answer:

The value of the definite integral $\int_{0}^{4} x e^{\frac{x}{3}} dx$ is approximately 20.381.

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