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Magazine Chapter 5: Problem 28
Find each indefinite integral. \(\int x^{2}(8 x+3) d x\)
Short Answer
Expert verified
\( 2x^4 + x^3 + C \)
Step by step solution
01
Expand the Expression Inside the Integral
First, we need to simplify the integrand. Expand the given expression \( x^2(8x + 3) \) using the distributive property. This gives us: \( x^2 \cdot 8x + x^2 \cdot 3 = 8x^3 + 3x^2 \).
02
Write the Expanded Integral
Now that we have expanded the expression, we can rewrite the original integral as follows: \( \int (8x^3 + 3x^2) \, dx \).
03
Integrate Each Term Separately
We will integrate each term individually. The integral of a term in the form \( ax^n \) is \( \frac{a}{n+1}x^{n+1} \). So, for each term:- For \( 8x^3 \), the integral is \( \frac{8}{4}x^4 = 2x^4 \).- For \( 3x^2 \), the integral is \( \frac{3}{3}x^3 = x^3 \).
04
Combine the Integrated Terms and Include the Constant of Integration
Now add the integrals of each term together: \( 2x^4 + x^3 \). Don't forget to add the constant of integration \( C \), which we include with indefinite integrals. The final result is: \( 2x^4 + x^3 + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
When faced with an integral like \( \int x^{2}(8x + 3) \, dx \), choosing the right integration technique is crucial. Here, the task is to integrate a polynomial expression, and the key step is to simplify the expression first. By expanding \( x^2(8x + 3) \) using the distributive property, we transform the problem into a simpler form: \( 8x^3 + 3x^2 \).
Once expanded, one of the most straightforward integration techniques, known as **term-by-term integration**, comes into play. This means integrating each term of the polynomial separately.
Once expanded, one of the most straightforward integration techniques, known as **term-by-term integration**, comes into play. This means integrating each term of the polynomial separately.
- Identify simpler terms in the expanded expression: here, \( 8x^3 \) and \( 3x^2 \).
- Integrate terms individually using the basic rule for integrating powers of \( x \): \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Polynomial Integration
Polynomial integration is a significant technique when dealing with integrals of polynomial functions, such as \( \int (8x^3 + 3x^2) \, dx \). This approach focuses on integrating each term of the polynomial separately to simplify the integration process.
Here's how polynomial integration works, in detail:
Here's how polynomial integration works, in detail:
- The expression is initially expanded, simplifying it by distributing terms appropriately, as seen with \( 8x^3 + 3x^2 \).
- Apply the integration power rule: for any term in the form of \( ax^n \), the integral is calculated as \( \frac{a}{n+1}x^{n+1} \).
Constant of Integration
When you calculate an indefinite integral, like \( \int (8x^3 + 3x^2) \, dx \), it鈥檚 crucial to remember the **constant of integration** \( C \). This constant represents an infinite number of possible constants that could be added to the antiderivative when evaluating indefinite integrals.
This element has an essential role:
This element has an essential role:
- Signifies the family of functions to which the antiderivative belongs.
- Compensates for any constant that was lost during differentiation, as derivatives of constants are zero.
- Ensures that all potential solutions to a differential equation are covered.
