Problem 36 - Applied Calculus

Question

Find each indefinite integral. \(\int(3 s+1)(3 s-1) d s\)

Step-by-Step Solution

Answer

The indefinite integral is \( 3s^3 - s + C \).

Step 1: Expand the Product

The given integral is \( \int (3s + 1)(3s - 1) \, ds \). First, expand the product using the distributive property: \((3s + 1)(3s - 1) = (3s)^2 - (1)^2 = 9s^2 - 1 \). So we rewrite the integral as \( \int (9s^2 - 1) \, ds \).

Step 2: Integrate Term by Term

Next, integrate each term individually. Break it down: \( \int 9s^2 \, ds \) and \( \int -1 \, ds \).1. For \( \int 9s^2 \, ds \), use the power rule: \( (9/3) s^3 = 3s^3 \).2. For \( \int -1 \, ds \), the integral is \(-s\).

Step 3: Combine the Integrals

Combine the results from the previous step: \( 3s^3 - s + C \), where \( C \) is the constant of integration.

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle used in various calculations. It states that the product of a number and a sum is the sum of the individual products. In simpler terms, when you have an expression like \(a(b + c)\), it can be expanded to \(ab + ac\). This property ensures that multiplication is spread out or 'distributed' across the terms added together.

In our exercise, we use the distributive property to expand the expression \( (3s + 1)(3s - 1) \). By applying the property here, we calculate it as \( 3s(3s - 1) + 1(3s - 1) \). Breaking it down further, we get \(9s^2 - 3s + 3s - 1\), which simplifies to \(9s^2 - 1\). This step prepares the expression for easier integration by breaking it into simpler parts.

Power Rule Integration

The power rule for integration is a handy technique for finding the integral of polynomials. This rule states that if you're integrating a term of the form \(x^n\), where \(n\) is not equal to -1, the result will be \((x^{n+1})/(n+1)\). Simply put, you add one to the exponent and divide by the new exponent. This rule greatly simplifies the process of integration for polynomials.

For our exercise specifically, we encounter the term \(9s^2\). Using the power rule, you increase the exponent by one, making it \(3\), and then divide by this new exponent: \((9/3)s^3 = 3s^3\).

• For a constant term, such as \(-1\), the rule is slightly different because there is essentially a hidden \(-1 imes s^0\). The integral of a constant \(-1\) with respect to \(s\) is \(-s\).

Using these rules makes the complex problem of integration surprisingly manageable.

Constant of Integration

The constant of integration, denoted by \(C\), is a crucial aspect of indefinite integrals. When you find the indefinite integral of a function, you essentially reverse the process of differentiation. However, because differentiation of a constant is zero, any constant term present in the original function is lost during differentiation.

This means that when we integrate, there could have been any constant, positive, negative, or zero, originally in that function—we simply don't know what it was. Therefore, we add \(C\) to our result to account for all possible constants that might have existed before we took the derivative.

In our exercise, after integrating each term separately, we combine them to get the expression \(3s^3 - s\). To show completeness and to indicate the possibility of an unknown constant, we add the constant of integration, resulting in our final answer: \(3s^3 - s + C\).

Always remember: including \(C\) in your solution is essential to reflect every potential antiderivative.