Problem 15 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(8 x^{3}-3 x^{2}+2\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(2x^4 - x^3 + 2x + C\).

Step 1: Identify the Integral Components

We need to integrate the function \(8x^3 - 3x^2 + 2\) with respect to \(x\). The function can be broken down into three separate terms: \(8x^3\), \(-3x^2\), and \(2\).

Step 2: Apply the Power Rule of Integration

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. Apply this rule to each term separately.

Step 3: Integrate Each Term Separately

- For \(8x^3\), the integral is \(\int 8x^3 \, dx = 8 \cdot \frac{x^{4}}{4} = 2x^4\).- For \(-3x^2\), the integral is \(\int -3x^2 \, dx = -3 \cdot \frac{x^3}{3} = -x^3\).- For \(2\), the integral is \(\int 2 \, dx = 2x\).

Step 4: Add the Integrated Terms

Combine the integrated terms and add the constant of integration \(C\). Thus,\[ \int (8x^3 - 3x^2 + 2) \, dx = 2x^4 - x^3 + 2x + C \]

Step 5: Final Check

Recheck each term to ensure the power rule was correctly applied and all components are accounted for, including the constant of integration. The integration is complete now.

Key Concepts

Integration Techniques

Integration is a crucial process used in calculus to find the area under a curve, among many other applications. There are several integration techniques that can be employed depending on the form of the function you are dealing with.

Some of the most commonly used methods include:

• Substitution: Useful when a substitution can simplify the integral into a more recognizable form.
• Integration by Parts: Often applied for products of functions, using the formula \(\int u \, dv = uv - \int v \, du\).
• Partial Fraction Decomposition: Perfect for integrating rational functions broken down into simpler fractions.
• Power Rule: As seen in our problem, simplifies powers of \(x\) by incrementally increasing their exponent.

When faced with an integration problem, choosing the right technique can dramatically simplify the process. In our original exercise, we use the power rule, a direct and efficient choice for polynomial functions.

Power Rule for Integration

The Power Rule for Integration is foundational in calculus, especially when working with polynomials. It enables you to integrate functions of the form \(x^n\) easily.

The rule states:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Here, \(n\) represents the exponent of \(x\), and \(C\) is the constant of integration, reflecting an indefinite integral. Each term of a polynomial can be integrated separately using this rule, as demonstrated in the original step-by-step solution.

By applying this to each component separately, you retain the structure of the polynomial while adjusting each term's power, making it a straightforward yet powerful method. This technique is perfect for students to master, as it forms the base for solving more complex integrals.

Mathematical Problem Solving

Effective mathematical problem solving involves breaking down complex problems into manageable parts, a strategy that was applied in the example. Let's look into the approach:

• Understand the Problem: Before diving in, make sure each term of the expression is clear. This is the assessment phase, where you outline what you're integrating and with respect to which variable.
• Choose the Appropriate Method: Selecting the right integration technique, like the power rule for polynomials, can simplify the process significantly.
• Step-by-Step Application: Work through each component of the expression one at a time. Keep your work organized and check each calculation thoroughly.
• Check Your Work: Always verify by differentiating your result. If you obtain the original function, your integration is likely correct. This verification step ensures accuracy and boosts confidence in your solution.

Approaching mathematical problems systematically, with clear steps and logic, helps ensure accuracy and develop problem-solving skills that are transferable to various types of mathematical challenges.