Problem 34 - Applied Calculus

Question

Find each indefinite integral. \(\int(x+5)(x-3) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(\frac{x^3}{3} + x^2 - 15x + C\).

Step 1: Expand the Integrand

First, expand the expression \((x+5)(x-3)\) by distributing the terms:\(x \cdot x + x \cdot (-3) + 5 \cdot x + 5 \cdot (-3)\).This simplifies to \(x^2 - 3x + 5x - 15\).Combine like terms to get the expanded form: \(x^2 + 2x - 15\).

Step 2: Integrate Each Term

Now, find the indefinite integral of the expanded polynomial, term by term:The integral of \(x^2\) is \(\frac{x^3}{3}\).The integral of \(2x\) is \(x^2\).The integral of \(-15\) is \(-15x\).Don't forget the constant of integration, \(C\).

Step 3: Combine the Integrated Terms

Combine the results of the term-by-term integration:\(\frac{x^3}{3} + x^2 - 15x + C\).This is the indefinite integral of the given function.

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of breaking down a product of expressions into a sum of simpler terms. In the exercise, the expression \((x+5)(x-3)\) is expanded using distribution. This involves multiplying each term in the first polynomial by every term in the second polynomial:

• Multiply the \(x\) from \((x+5)\) by \(x\) from \((x-3)\), which equals \(x^2\).
• Multiply \(x\) by \(-3\), resulting in \(-3x\).
• Next, multiply \(5\) by \(x\), which gives \(5x\).
• Finally, multiply \(5\) by \(-3\), resulting in \(-15\).

Combine these terms to obtain the expanded polynomial: \(x^2 + 2x - 15\). This step simplifies the integrand, making it easier to integrate term by term.

Integration Techniques

Integration is the process of finding a function's antiderivative. For polynomials, this involves reversing the process of differentiation, term by term. Let's look at the integration of the expanded polynomial \(x^2 + 2x - 15\). Each term is integrated individually:

• The integral of \(x^2\) is found using the power rule: increase the exponent by one and divide by the new exponent. Thus, \(\int x^2 dx = \frac{x^3}{3}\).
• For the linear term \(2x\), apply the power rule to get \(\int 2x \, dx = x^2\), because \(\int x \, dx = \frac{x^2}{2}\), and multiplying by \(2\) gives \(x^2\).
• The constant term \(-15\) integrates to \(-15x\) because the integral of a constant \(c\) is \(cx\).

By integrating each term separately and then combining them, we arrive at the solution: \(\frac{x^3}{3} + x^2 - 15x + C\).

Constant of Integration

After finding the indefinite integral, it's essential to include the constant of integration, represented by \(C\). This constant accounts for the fact that antiderivatives are not unique. Each has infinite versions differing only by a constant.

• If you differentiate a function, any constant term disappears (since the derivative of a constant is zero).
• When finding the antiderivative, we must account for this "lost" constant by adding \(C\).
• Thus, the general solution to each indefinite integral includes this constant to represent all possible antiderivatives.

In the exercise solution, including \(C\) is crucial. It ensures you have the most general form of the antiderivative: \(\frac{x^3}{3} + x^2 - 15x + C\). This reflects all functions differentiating to the original integrand.