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Chapter 25: Problem 16

Integrate each of the given expressions. \(\int x(x-2)^{2} d x\)

Short Answer

Expert verified
\(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2 + C\)

Step by step solution

01

Expand the Expression

Start by expanding the expression within the integral. The integrand is \(x(x-2)^2\). First, expand \((x-2)^2\) to get \(x^2 - 4x + 4\). Therefore, the expression becomes \(x(x^2 - 4x + 4)\).
02

Distribute and Simplify

Distribute the \(x\) across the terms inside the parentheses: \(x \cdot x^2 - x \cdot 4x + x \cdot 4\), resulting in \(x^3 - 4x^2 + 4x\). Now, the integral is \(\int (x^3 - 4x^2 + 4x) \, dx\).
03

Integrate Term by Term

Integrate each term separately: 1. The integral of \(x^3\) is \(\frac{x^4}{4}\).2. The integral of \(-4x^2\) is \(-\frac{4x^3}{3}\).3. The integral of \(4x\) is \(2x^2\).Thus, the integrated expression is \(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2\).
04

Include Constant of Integration

Don't forget to add the constant of integration, \(C\), as the last step in indefinite integrals. So, the final answer is \(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2 + C\).

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Key Concepts

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

Expansion of Algebraic Expressions
When you encounter an algebraic expression that needs to be integrated, it's often beneficial to expand it first. This simplifies the integration process. In our exercise, the expression \(x(x-2)^2\) needs to be expanded before integrating. To do this:
  • First, expand \((x-2)^2\) into \(x^2 - 4x + 4\) by squaring the binomial.
  • Then, distribute \(x\) across the expanded terms: \(x(x^2 - 4x + 4)\), resulting in \(x^3 - 4x^2 + 4x\).

The expansion allows each term to be individually integrated in the following steps. Expanding expressions breaks down complex formulas into simpler parts, making the solution more approachable.
Term-by-Term Integration
Once the expression is expanded, each term can be integrated separately. This is known as term-by-term integration. Here's how it works for the expression \(x^3 - 4x^2 + 4x\):
  • Integrate \(x^3\) to get \(\frac{x^4}{4}\).
  • Integrate \(-4x^2\) to get \(-\frac{4x^3}{3}\).
  • Integrate \(4x\) to get \(2x^2\).

Each term is integrated according to the power rule, which states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\) where \(neq -1\). This method simplifies the process, focusing on one term at a time, ensuring accuracy in calculations.
Constant of Integration
After performing term-by-term integration, an important final step is to add the constant of integration, denoted by \(C\). This constant reflects that there are infinitely many antiderivatives for a function.

When integrating functions, especially indefinite integrals like our example, adding this constant is crucial. It accounts for all possible vertical shifts of the antiderivative graph. In our solution:
  • After integrating, we reach the expression \(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2\).
  • Adding \(C\) gives the complete solution: \(\frac{x^4}{4} - \frac{4x^3}{3} + 2x^2 + C\).

Neglecting the constant \(C\) can lead to incomplete solutions and misunderstandings in problems involving indefinite integrals.

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