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Chapter 5: Problem 16

Find each indefinite integral. \(\int\left(12 x^{3}+3 x^{2}-5\right) d x\)

Short Answer

Expert verified
The indefinite integral is \(3x^4 + x^3 - 5x + C\).

Step by step solution

01

Identify the Polynomials

The given integral is a polynomial expression: \(12x^3 + 3x^2 - 5\). We need to integrate each term separately.
02

Use the Power Rule for Integration

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration. Apply this to each term: \(12x^3\), \(3x^2\), and \(-5\).
03

Integrate Each Term Separately

- For \(12x^3\), the integral is \(\int 12x^3 \, dx = 12 \cdot \frac{x^{4}}{4} = 3x^4\).- For \(3x^2\), the integral is \(\int 3x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = x^3\).- For \(-5\), the integral is \(\int -5 \, dx = -5x\).
04

Combine the Integrated Terms and Add the Constant of Integration

Combine all the integrated terms: \(3x^4 + x^3 - 5x\). Don鈥檛 forget to add the constant of integration, \(C\), to get the final answer: \(3x^4 + x^3 - 5x + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Power Rule for Integration
The power rule for integration is a fundamental tool when working with polynomial functions. It helps us integrate expressions of the form \(x^n\), where \(n\) is any real number. This rule states that:
  • If \(n eq -1\), then \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).
  • This formula allows you to raise the power of \(x\) by one and divide by the new power.
Applying this rule to each term in a polynomial helps break down complex expressions into easier, manageable pieces for integration.
For instance, if you have a term like \(12x^3\), using the power rule gives us \(12 \cdot \frac{x^{4}}{4} = 3x^4\). By following this method, we can systematically handle each term in a polynomial.
Executing Polynomial Integration
Polynomial integration involves integrating each term of a polynomial separately using rules like the power rule. Polynomials are expressions consisting of terms in the form \(ax^n\). Here's how we integrate a polynomial step-by-step:
Take each term independently, apply known integration rules like the power rule, and calculate its integral.
  • For example, consider the polynomial \(12x^3 + 3x^2 - 5\).
  • Integrate \(12x^3\): The result is \(3x^4\).
  • Integrate \(3x^2\): Here, the result is \(x^3\).
  • Integrate the constant term \(-5\): It becomes \(-5x\).
By transforming each term, polynomial integration becomes a straightforward task, allowing us to integrate complex expressions efficiently.
Adding the Constant of Integration
When solving indefinite integrals, it's crucial to remember the constant of integration, often denoted as \(C\). This constant accounts for any constant value that could have been differentiated to result in the original function:
The role of \(C\) is to represent all possible antiderivatives of a function.
  • Since differentiation of a constant is zero, the original function may have an "invisible" constant whose derivative was zero.
  • By adding \(C\), you acknowledge that multiple functions could provide the same derivative.
In the example \(3x^4 + x^3 - 5x + C\), \(C\) ensures that all potential solutions to the indefinite integral are covered, reflecting the inherent nature of indefinite integration.

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Most popular questions from this chapter

These exercises review material that will be helpful in Section \(5.6\). Find the derivative of each function. \(\ln \left(x^{3}-1\right)\)

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