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Chapter 5: Problem 27

Find each indefinite integral. $$ \int 12 x^{2}(x-1) d x $$

Short Answer

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

Step by step solution

01

Expand the Expression

First, we need to expand the expression inside the integral. The expression is \(12x^2(x-1)\). Distribute \(12x^2\) to \(x\) and \(-1\): \(12x^2 \times x = 12x^3\) \(12x^2 \times (-1) = -12x^2\)This results in the expression \(12x^3 - 12x^2\).
02

Write the Integral of the Expanded Expression

Substitute the expanded expression into the integral. The integral becomes: \[\int (12x^3 - 12x^2) \, dx\]
03

Integrate Term by Term

Integrate each term separately. For the first term, \(\int 12x^3 \, dx\), we use the power rule: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]For \(12x^3\), the integral is \(\frac{12x^4}{4} = 3x^4\).For the second term, \(\int -12x^2 \, dx\), similarly: \(-12 \times \int x^2 \, dx = -12 \times \frac{x^3}{3} = -4x^3\).
04

Combine the Integrated Terms and Add Constant of Integration

Combine the results of the integration:\(3x^4 - 4x^3\).Since it's an indefinite integral, add the constant of integration \(+ C\):\[3x^4 - 4x^3 + C\]

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Key Concepts

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

Power Rule of Integration
The power rule of integration is a fundamental technique used to simplify the integration of polynomial functions. It is based on applying a specific formula to each term:
  • The formula is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
  • We'll integrate each term of the polynomial separately using this rule.
In our example integral of \( \int (12x^3 - 12x^2) \, dx \):
  • For the term \( 12x^3 \), apply the power rule by increasing the exponent by one (from 3 to 4), and dividing by the new exponent. Hence, \( \int 12x^3 \, dx = 3x^4 \).
  • For \( -12x^2 \), use the same approach: the exponent becomes 3, and so \( -12 \times \frac{x^3}{3} = -4x^3 \).
Remember to adjust the coefficients appropriately to get the correct result. The power rule is essential for efficiently finding antiderivatives of polynomial expressions.
Polynomial Expansion
Polynomial expansion involves breaking down expressions to simplify the process of integration or differentiation. In many cases, the target polynomial expression can be expanded to give a clearer view of the individual terms.
  • Our given example \( 12x^2(x-1) \) demonstrates an essential use of this technique.
  • You distribute \( 12x^2 \) over both terms in the parenthesis: \( 12x^2 \times x\) and \( 12x^2 \times -1 \), resulting in \( 12x^3 - 12x^2 \).
This straightforward expansion ensures each term within the expression can be individually approached using methods such as the power rule of integration. Expanding the polynomial prepares the expression for term-wise operations, making subsequent integration much clearer and manageable.
Constant of Integration
When dealing with indefinite integrals, don't forget the constant of integration, represented as \( + C \). This constant accounts for the fact that integrating a function results in a family of functions, all differing by a constant.
  • In indefinite integrals, since there's no specific limit defined, the result is not a single function but a general form.
  • The constant \( C \) ensures that all possible functions derived from the integral are represented.
For the integral \( \int (12x^3 - 12x^2) \, dx \), after performing the integration we found: \( 3x^4 - 4x^3 \).
  • You append \( + C \) to represent all potential vertical shifts of the antiderivative.
Always include the constant of integration for indefinite integrals to accurately reflect the full set of possible solutions available.

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