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Magazine Chapter 5: Problem 40
Find each indefinite integral. \(\int(t-1)^{3} d t\)
Short Answer
Expert verified
\(\frac{t^4}{4} - t^3 + \frac{3t^2}{2} - t + C\)
Step by step solution
01
Identify the Integral Form
We are given the integral \( \int (t-1)^3 \, dt \). This is a polynomial expression in a simple form and can be integrated using the power rule for integration.
02
Use Substitution if Necessary
Since the expression \((t-1)^3\) is a simple polynomial form that does not require substitution for evaluation at this point, we can directly integrate it by expanding it if preferred or directly using the power rule.
03
Expand the Expression
First, expand \((t-1)^3\). This can be done using the binomial expansion:\[(t-1)^3 = t^3 - 3t^2 + 3t - 1\]This step prepares it for direct integration term by term.
04
Integrate Term by Term
Now integrate each term of the expanded expression:\[\int (t^3 - 3t^2 + 3t - 1) \, dt = \int t^3 \, dt - \int 3t^2 \, dt + \int 3t \, dt - \int 1 \, dt\]Using the power rule for each term:\[\int t^3 \, dt = \frac{t^4}{4}, \int 3t^2 \, dt = \frac{3t^3}{3} = t^3,\int 3t \, dt = \frac{3t^2}{2},\int 1 \, dt = t\]So, the integrated form is:\[\frac{t^4}{4} - t^3 + \frac{3t^2}{2} - t\]
05
Combine the Integrated Terms
Combine the results from each individual term from the previous step:\[\int (t-1)^3 \, dt = \frac{t^4}{4} - t^3 + \frac{3t^2}{2} - t + C\]where \(C\) is the constant of integration, as this is an indefinite integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule for Integration
The power rule for integration is a fundamental concept for solving indefinite integrals, especially with polynomials. When you encounter expressions like \(x^n\), the power rule allows us to find the antiderivative easily.
Here鈥檚 how it works:
When applying this rule, remember:
Here鈥檚 how it works:
- For any term \(x^n\), where \(n eq -1\), the integral is given by \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
When applying this rule, remember:
- The exponent increases by one.
- You divide by the new exponent.
Polynomial Integration
Polynomial integration involves integrating expressions that are polynomials. A polynomial is a mathematical expression consisting of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents.
For example, in the given problem, the polynomial \( (t-1)^3 \) was expanded to \( t^3 - 3t^2 + 3t - 1 \). To integrate it:
This makes the integration process straightforward. Each term is individually simpler to integrate than the entire polynomial. Remember, each term鈥檚 coefficient affects the result, and be sure to include the constant of integration and any negative signs.
Integration here doesn鈥檛 just sum these terms but reconstructs them into the integrated polynomial, representing the family of all possible functions that could yield the original polynomial after differentiation.
For example, in the given problem, the polynomial \( (t-1)^3 \) was expanded to \( t^3 - 3t^2 + 3t - 1 \). To integrate it:
- Expand each term.
- Apply the power rule to each part separately.
This makes the integration process straightforward. Each term is individually simpler to integrate than the entire polynomial. Remember, each term鈥檚 coefficient affects the result, and be sure to include the constant of integration and any negative signs.
Integration here doesn鈥檛 just sum these terms but reconstructs them into the integrated polynomial, representing the family of all possible functions that could yield the original polynomial after differentiation.
Binomial Expansion
In mathematics, the binomial expansion is a convenient way to expand expressions raised to a power. For a binomial expression like \( (a + b)^n \), it means finding the expanded form without manually multiplying each pair.
This expansion uses the formula: \ [(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k].\
In the problem, \( (t-1)^3 \) was expanded using binomial expansion. This resulted in the terms \(t^3 - 3t^2 + 3t - 1\), providing a straightforward form for integration.
By using binomial expansion, we can greatly simplify complex expressions, making subsequent operations like integration much more manageable.
This expansion uses the formula: \ [(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k].\
- \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), is the binomial coefficient, representing combinations choosing \(k\) elements from \(n\).
In the problem, \( (t-1)^3 \) was expanded using binomial expansion. This resulted in the terms \(t^3 - 3t^2 + 3t - 1\), providing a straightforward form for integration.
By using binomial expansion, we can greatly simplify complex expressions, making subsequent operations like integration much more manageable.
