Problem 54 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int(x-1)^{2} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(\frac{x^3}{3} - x^2 + x + C\).

Step 1: Expand the Expression

First, expand the expression \((x - 1)^2\) using the binomial formula. This gives: \((x - 1)^2 = x^2 - 2x + 1\).

Step 2: Separate the Integrals

The integral \(\int (x^2 - 2x + 1) \, dx\) can be split into separate integrals: \[ \int x^2 \, dx - 2 \int x \, dx + \int 1 \, dx \]

Step 3: Integrate Each Term Individually

Now, integrate each term: 1. \(\int x^2 \, dx = \frac{x^3}{3} + C_1\) 2. \(-2 \int x \, dx = -2 \times \frac{x^2}{2} + C_2 = -x^2 + C_2\) 3. \(\int 1 \, dx = x + C_3\)

Step 4: Combine the Results

Combine the results of the integrated terms to get the final indefinite integral expression: \[ \frac{x^3}{3} - x^2 + x + C \] where \(C = C_1 + C_2 + C_3\) is the constant of integration.

Key Concepts

Understanding the Binomial Formula

When dealing with expressions like \((x - 1)^2\), the binomial formula becomes a handy tool to simplify or expand them. The binomial formula helps in breaking down squared or higher power polynomial expressions into a sum of terms. For two terms, \((a + b)^n\), the expansion is given by the formula:

• \((a + b)^n = \sum_{{k=0}}^{n} \binom{n}{k} a^{n-k} b^{k}\)

In our exercise, \((x - 1)^2\) can be expanded by applying the simple case of the binomial theorem where \(n = 2\). Expanding gives us \(x^2 - 2x + 1\). This expanded form is straightforward yet critical, as it sets the stage for easier integration in subsequent steps.

Exploring Integration Techniques

Integration is a core operation in calculus, used to find areas under curves or solve differential equations. In this exercise, we approached the problem by decomposing the original integral into simpler parts to handle each separately.
This method greatly simplifies the process and makes the integration of complex expressions manageable. Let’s break down each part:

• The original integral is \(\int(x^2 - 2x + 1)\, dx\).
• We then split it into three separate integrals: \(\int x^2 \, dx\), \(-2 \int x \, dx\), and \(\int 1 \, dx\).
• This separation allows us to use the basic power rule of integration, where \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).

Each term is integrated individually, allowing us to combine them later to yield a final comprehensive solution. Each small step leads us closer to understanding the change that the function undergoes, providing insight into its cumulative behavior over an interval.

The Importance of the Constant of Integration

In indefinite integrals, the constant of integration \(C\) plays a crucial role. The idea here is that when you integrate a function, there is an infinite number of antiderivatives. Thus, this constant is added to represent that infinity of possible vertical shifts that the original function could have undergone.
Here’s how it works:

• After integrating, say \(\int x^2 \, dx\), the result is \(\frac{x^3}{3} + C_1\) where \(C_1\) is a constant.
• Similarly, after solving the complete integral, you combine all constants: \(C = C_1 + C_2 + C_3\).

The constant of integration may seem minor, but it's essential for ensuring that every antiderivative is accounted for. In any real-world problem, this constant can mean different initial values or conditions that materially affect outcomes. It emphasizes the significance of defining your specific context in which the mathematical solution is applied.