Problem 54 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(\int(x-1)^2 \, dx = \frac{x^3}{3} - x^2 + x + C\)

Step 1: Expand the integrand

Start by expanding the expression \((x-1)^2\) before integrating. \[(x-1)^2 = x^2 - 2x + 1\]

Step 2: Rewrite the integral

Substitute the expanded expression back into the integral. \[\int(x-1)^{2} \, dx = \int(x^2 - 2x + 1) \, dx\]

Step 3: Integrate term by term

Integrate each term separately. The integral of \(x^2\) is \(\frac{x^3}{3}\), the integral of \(-2x\) is \(-x^2\), and the integral of \(1\) is \(x\).\[\int(x^2 - 2x + 1) \, dx = \frac{x^3}{3} - x^2 + x + C\]

Step 4: Collect the results

Combine all the terms and simplify if necessary. The result of the indefinite integral is:\[\frac{x^3}{3} - x^2 + x + C\]

Key Concepts

Integration Techniques

When dealing with indefinite integrals, it is essential to choose the correct integration technique to simplify the process. One of the key techniques is recognizing when to expand an expression for easier integration. For instance, if you have an expression such as \((x-1)^2\), direct integration may not be straightforward. Expanding the expression results in a polynomial, which is much easier to handle. This specific technique falls under the broader category of manipulating algebraic expressions to make integration simpler.It's also important to remember that each integration technique serves a purpose. By understanding and knowing when to expand, or simplify an expression, you lay the foundation for tackling more complex integrals in calculus.

Polynomial Expansion

Polynomial expansion transforms complicated expressions into simpler polynomial form. Let's take a look at how this works with our given integral \((x-1)^2\).- Begin with the expression \((x-1)^2\). Use the distributive property to expand it to a polynomial.- This results in the expanded form: \(x^2 - 2x + 1\).By expanding polynomials like this, you make terms individually recognizable, simplifying the process of integration. Once you have the expression in expanded polynomial form, each term can then be integrated separately. This is crucial because polynomials are among the simplest functions to integrate.
Expanding polynomials is often the first step in preparing expressions for integration, making otherwise challenging calculus problems elegantly solvable!

Calculus Steps

Calculus takes a systematic approach to solve mathematical problems, especially in integration. Let’s break it down using our example of \(\int(x-1)^2 \, dx\).- **Step 1**: **Expand the Expression** - Begin with expanding \((x-1)^2\) to get \(x^2 - 2x + 1\).
This step sets the stage for easier handling of the integral.- **Step 2**: **Substitute Back into Integral** - Replace \((x-1)^2\) with its expanded form in the integral.
This gives: \(\int(x^2 - 2x + 1) \, dx\).- **Step 3**: **Integrate Term by Term** - Integrate each term separately:

• \(\int x^2 \, dx = \frac{x^3}{3}\)
• \(\int -2x \, dx = -x^2\)
• \(\int 1 \, dx = x\)

- **Step 4**: **Combine the Results** - Collect the integrated terms to form the final result:
This outputs \(\frac{x^3}{3} - x^2 + x + C\).This sequence of actions in calculus allows us to break down complex processes into understandable and manageable steps, guiding us to find the indefinite integral.