[x²√√x + 17 dx

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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State the most appropriate technique of integration to utilize for this problem, then evaluate the integral.
### Calculus: Indefinite Integral of a Polynomial Function with a Radical Expression

The integral to be evaluated is:

\[ \int x^2 \sqrt{x} + 17 \, dx \]

Let's break down the integral into simpler components for better understanding:

1. **Expression Breakdown**:
    - The integral consists of two main parts: \(x^2 \sqrt{x}\) and \(17\).
    - The variable of integration is \(x\).

2. **Simplification Step**:
    - Note that \(\sqrt{x}\) can be rewritten as \(x^{1/2}\).
    - Thus, \(x^2 \sqrt{x} = x^2 \cdot x^{1/2} = x^{2 + 1/2} = x^{5/2}\).

3. **Integral Representation**:
    - The integral can now be written as:
      \[ \int x^{5/2} + 17 \, dx \]

4. **Integration Process**:
    - Integrate each term separately:
      \[ \int x^{5/2} \, dx + \int 17 \, dx \]
    - For \(\int x^{5/2} \, dx\):
      \[ \int x^{5/2} \, dx = \frac{x^{(5/2) + 1}}{(5/2) + 1} = \frac{x^{7/2}}{7/2} = \frac{2}{7} x^{7/2} \]
    - For \(\int 17 \, dx\):
      \[ \int 17 \, dx = 17x \]

5. **Combining the Results**:
    - Add the results from both integrals:
      \[ \frac{2}{7} x^{7/2} + 17x + C \]
    - Where \(C\) is the constant of integration.

6. **Final Answer**:
    - The indefinite integral is:
      \[ \frac{2}{7} x^{7/2} + 17x + C \]

This computation demonstrates the integration process of a polynomial term combined with a radical expression and helps reinforce the methods of calculus to solve such problems.
Transcribed Image Text:### Calculus: Indefinite Integral of a Polynomial Function with a Radical Expression The integral to be evaluated is: \[ \int x^2 \sqrt{x} + 17 \, dx \] Let's break down the integral into simpler components for better understanding: 1. **Expression Breakdown**: - The integral consists of two main parts: \(x^2 \sqrt{x}\) and \(17\). - The variable of integration is \(x\). 2. **Simplification Step**: - Note that \(\sqrt{x}\) can be rewritten as \(x^{1/2}\). - Thus, \(x^2 \sqrt{x} = x^2 \cdot x^{1/2} = x^{2 + 1/2} = x^{5/2}\). 3. **Integral Representation**: - The integral can now be written as: \[ \int x^{5/2} + 17 \, dx \] 4. **Integration Process**: - Integrate each term separately: \[ \int x^{5/2} \, dx + \int 17 \, dx \] - For \(\int x^{5/2} \, dx\): \[ \int x^{5/2} \, dx = \frac{x^{(5/2) + 1}}{(5/2) + 1} = \frac{x^{7/2}}{7/2} = \frac{2}{7} x^{7/2} \] - For \(\int 17 \, dx\): \[ \int 17 \, dx = 17x \] 5. **Combining the Results**: - Add the results from both integrals: \[ \frac{2}{7} x^{7/2} + 17x + C \] - Where \(C\) is the constant of integration. 6. **Final Answer**: - The indefinite integral is: \[ \frac{2}{7} x^{7/2} + 17x + C \] This computation demonstrates the integration process of a polynomial term combined with a radical expression and helps reinforce the methods of calculus to solve such problems.
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