Problem 13 - Applied Calculus

Question

Find each indefinite integral. \(\int(8 x-5) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(4x^2 - 5x + C\).

Step 1: Identify the Integral Components

The given integral is \(\int (8x - 5) \, dx\). We recognize that this integral involves the sum of two terms: \( 8x \) and \(-5\).

Step 2: Apply the Sum Rule of Integration

Using the sum rule for integrals, we can split the integral into two separate integrals: \( \int 8x \, dx - \int 5 \, dx \). This allows us to integrate each term independently.

Step 3: Integrate Each Term Separately

First, integrate \(8x\). The integral of \(x\) is \(\frac{x^2}{2}\), so \(\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2\). Next, integrate \(-5\). The integral of a constant \(c\) is \(cx\), so \(\int -5 \, dx = -5x\).

Step 4: Combine the Integrated Terms

Combine the results of the individual integrations: \(4x^2 - 5x\).

Step 5: Add the Constant of Integration

Remember that indefinite integrals include a constant of integration. Thus, the final result is \(4x^2 - 5x + C\), where \(C\) is the constant of integration.

Key Concepts

Sum Rule of Integration

When dealing with indefinite integrals, it's essential to understand the sum rule of integration. This rule can simplify the process significantly. If you have the integral of a sum, like \(\int (8x - 5) \, dx\), you can separate it into two smaller, more manageable integrals. This method is called the sum rule.Here's how it works:

• Separate the integral into individual terms: \(\int 8x \, dx - \int 5 \, dx\).
• This way, you can focus on integrating simple elements.

The primary advantage of the sum rule is that it allows you to tackle each part of the integrand individually, making complex expressions easier to manage.

Integration of Polynomial Functions

Integrating polynomial functions is a fundamental skill in calculus. The example given, \(\int 8x \, dx\), illustrates how to integrate such expressions.Here are the basic steps to integrate a polynomial:

• Identify the term with a variable, such as \(8x\).
• Use the power rule for integration: to integrate \(x^n\), you add one to the exponent, then divide by the new exponent: \(\frac{x^{n+1}}{n+1}\).
• For \(8x\), this means: \(8 \cdot \frac{x^2}{2} = 4x^2\).

By understanding these steps, you can integrate any polynomial function with confidence, turning the process into a straightforward calculation.

Constant of Integration

When working with indefinite integrals, always remember to add the constant of integration, represented as \(C\). Here's why the constant is important:

• Indefinite integrals represent a family of functions, each differing by a constant.
• Without \(C\), you wouldn't account for all possible functions that could differentiate to the original integrand.
• For the integral we solved, \(4x^2 - 5x\), the full solution is actually \(4x^2 - 5x + C\).

Adding \(C\) ensures your solution is complete, reflecting all possible values the function could take. It's an essential part of expressing indefinite integrals properly.