Problem 27 - Applied Calculus

Question

Find each indefinite integral. \(\int 12 x^{2}(x-1) d x\)

Step-by-Step Solution

Answer

\(3x^4 - 4x^3 + C\)

Step 1: Simplify the Expression

Before integrating, simplify the expression by expanding the integrand. Use distribution to expand the expression: \(12x^2(x - 1) = 12x^3 - 12x^2\). Thus, our integral becomes: \(\int (12x^3 - 12x^2) \, dx\).

Step 2: Integrate Each Term Separately

Now, integrate each term in the expanded expression separately. Start with the term \(12x^3\): \(\int 12x^3 \, dx = \frac{12}{4}x^4 = 3x^4\).Next, integrate \(-12x^2\):\(\int -12x^2 \, dx = -\frac{12}{3}x^3 = -4x^3\).

Step 3: Combine the Integrated Terms

Combine the results of each integrated term and add the constant of integration \(C\):The integral of \(\int 12x^3 - 12x^2 \, dx\) is:\(3x^4 - 4x^3 + C\).

Key Concepts

Integration Techniques

When tackling indefinite integrals, a crucial first step is to apply effective integration techniques. Let's begin by understanding that integration is the reverse operation of differentiation. We seek a function whose derivative matches the integrand. While several techniques exist, such as substitution and integration by parts, the most straightforward method for polynomials is direct integration. Often, it's helpful to first simplify the integrand if possible. This might involve expanding expressions, as seen in our example. By simplifying \(12x^2(x-1)\) to \(12x^3 - 12x^2\), we prepare the expression for smoother integration. Therefore, selecting the optimal technique involves assessing the complexity of the problem and simplifying where necessary.

Polynomial Integration

Polynomial integration involves integrating terms of the form \(ax^n\). Each term can be integrated by applying the power rule for integration. This rule states that for a polynomial term \(x^n\), the integral is \(\frac{x^{n+1}}{n+1}\). Consider the example term \(12x^3\):

• Apply the power rule: \(\int 12x^3 \, dx = \frac{12}{4}x^4 = 3x^4\).

Similarly, for the term \(-12x^2\):

• Apply the power rule: \(\int -12x^2 \, dx = -\frac{12}{3}x^3 = -4x^3\).

By integrating each term separately, we can then sum the results. Thus, the integral of the simplified expression becomes \(3x^4 - 4x^3 + C\), where \(C\) is the constant of integration.

Calculus Problem Solving

Solving calculus problems, especially indefinite integrals, often involves a strategic approach to break down the problem into manageable steps. For polynomial integration, this means identifying each term, applying the rules of integration separately, and then combining the results. Each calculus problem might vary slightly in approach based on the integrand's complexity. Ensuring clarity in each step helps maintain accuracy.Here's a quick problem-solving strategy for indefinite integrals:

• Start by simplifying the integrand if it's a product or quotient.
• Use integration rules like the power rule to find antiderivatives of straightforward terms.
• Add a constant of integration (\(C\)) to your solution, as indefinite integrals represent a family of functions.

Each of these steps aligns with systematic calculus problem-solving, thus making the integral easier to solve step-by-step.