Problem 26 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the integral. \(\int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(3 \ln 2 - 1\).

Step 1: Break Down the Integral

Given the integral \( \int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} d x \), we need to simplify the expression first. Rewrite the integrand \( \frac{(x-1)^{3}}{x^{2}} \) by expanding \((x-1)^3\) as \(x^3 - 3x^2 + 3x - 1\). This gives us: \(\frac{x^3}{x^2} - \frac{3x^2}{x^2} + \frac{3x}{x^2} - \frac{1}{x^2} = x - 3 + \frac{3}{x} - \frac{1}{x^2}\).

Step 2: Set Up the Integral with Separated Terms

Now, separate the integral into four separate terms for easier calculation. This yields:\[\int_{1}^{2} (x - 3 + \frac{3}{x} - \frac{1}{x^2}) \, dx = \int_{1}^{2} x \, dx - 3\int_{1}^{2} 1 \, dx + 3\int_{1}^{2} \frac{1}{x} \, dx - \int_{1}^{2} \frac{1}{x^2} \, dx\]

Step 3: Evaluate Each Term Individually

1. Evaluate \(\int_{1}^{2} x \, dx\): The integral of \(x\) is \(\frac{x^2}{2}\). So, compute from 1 to 2: \[\frac{2^2}{2} - \frac{1^2}{2} = 2 - \frac{1}{2} = \frac{3}{2}\] 2. Evaluate \(-3\int_{1}^{2} 1 \, dx\): This becomes \(-3[x]_1^2 = -3(2 - 1) = -3\) 3. Evaluate \(3\int_{1}^{2} \frac{1}{x} \, dx\): The integral of \(\frac{1}{x}\) is \(\ln|x|\). So, compute: \[3(\ln|2| - \ln|1|) = 3(\ln 2 - 0) = 3 \ln 2\] 4. Evaluate \(-\int_{1}^{2} \frac{1}{x^2} \, dx\): The integral of \(\frac{1}{x^2}\) is \(-\frac{1}{x}\). Thus, compute: \(-[-\frac{1}{x}]_1^2 = -\left(-\frac{1}{2} + 1\right) = \frac{1}{2}\)

Step 4: Combine the Results

Add all calculated parts from Step 3:\[\left(\frac{3}{2} - 3 + 3 \ln 2 + \frac{1}{2}\right)\]Simplify this expression:\[\frac{3}{2} + \frac{1}{2} - 3 + 3 \ln 2 = 2 - 3 + 3 \ln 2 = 3 \ln 2 - 1\]

Step 5: State the Final Result

The evaluated integral of \( \int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} dx \) is \(3 \ln 2 - 1\).

Key Concepts

Step by Step Integration

Understanding each step in the integration process is crucial for solving calculus problems. Let's break down the exercise. First, we're given the integral \( \int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} \, dx \). Our goal is to simplify the integrand, which is the expression inside the integral, \( \frac{(x-1)^{3}}{x^{2}} \).
By expanding \((x-1)^3\), we identify each term and divide them by \(x^2\) individually. This simplifies to the expression \(x - 3 + \frac{3}{x} - \frac{1}{x^2}\). Now, the integral can be rewritten with separate terms. This step-by-step simplification allows us to track down how each part of the expression influences the result. It's like solving a puzzle one piece at a time, ensuring precision every step of the way. Breaking down an integral makes it more manageable and helps in understanding the broader math concepts.

Definite Integrals

A definite integral, such as \( \int_{1}^{2} \), evaluates the area under a curve between two specified points on a graph. In our example, we determine the precise area under the function \( \frac{(x-1)^{3}}{x^{2}} \) from \( x = 1 \) to \( x = 2 \).
This kind of integral is fundamental in calculus as it provides practical results like calculating distances, areas, and volumes in real-world scenarios. When solving a definite integral, a crucial step is to apply the calculated antiderivative over the limits, which are 1 and 2 in our case. We essentially 'plug in' these boundaries to evaluate it, like substituting limits into a formula.
Calculating definite integrals involves a clear understanding of how boundaries influence a function's cumulative value, and skillfully applying it to harness the function's full potential.

Integration Techniques

Integration involves various techniques, each creating a toolbox for tackling different mathematical problems. For the given integral \( \int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} \, dx \), we used expansion and separation of terms.
Common techniques include:

• Substitution: Swapping parts of the integrand with variables to simplify calculations.
• Integration by parts: Splitting the product of functions into simpler parts.
• Partial fraction decomposition: Breaking rational expressions into simpler fractions.

For our example, we mainly expanded and separated the expressions since direct integration of the combined form wasn't feasible. Choosing the right technique simplifies the process and ensures accurate results. Mastering these techniques gives you the flexibility to tackle any integral you may encounter. Remember, each technique has its time and place, functioning as a solution-oriented strategy to unwind complex expressions.