Calculating The Indefinite Integral Of (5s + 6)^2

Introduction

In this article, we will delve into the process of determining the indefinite integral of the function (5s + 6)^2. Indefinite integrals, also known as antiderivatives, are a fundamental concept in calculus. They represent the family of functions whose derivative is equal to the given function. Mastering the techniques of indefinite integration is crucial for solving a wide range of problems in mathematics, physics, engineering, and other scientific disciplines.

Before we begin, let's briefly recap the basics of indefinite integration. The indefinite integral of a function f(x) is denoted by ∫f(x) dx, and it represents the general antiderivative of f(x). The result of indefinite integration is always a family of functions, differing only by a constant term, often denoted by C. This constant of integration arises because the derivative of a constant is always zero, so when we reverse the process of differentiation, we lose information about any constant term that may have been present in the original function.

The power rule of integration is a cornerstone technique. This rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is any real number except -1. We will use the power rule extensively in this process. Additionally, the constant multiple rule, ∫cf(x) dx = c∫f(x) dx, will be employed, where c is a constant. This allows us to move constant factors outside the integral sign, simplifying the integration process.

Our main task is to find the indefinite integral of (5s + 6)^2 with respect to s. This involves expanding the square, applying the power rule and constant multiple rule, and simplifying the result. The steps will be shown in detail to ensure clarity and comprehension. By the end of this article, you will have a clear understanding of how to solve this type of indefinite integral and be well-equipped to tackle similar problems.

Expanding the Integrand

To begin, let's address the integral ∫(5s + 6)^2 ds. The first step in finding this indefinite integral involves expanding the square of the binomial (5s + 6). This will allow us to rewrite the integrand in a more manageable form, making it easier to apply the power rule of integration. Expanding the square involves multiplying the binomial by itself: (5s + 6)^2 = (5s + 6)(5s + 6).

Using the distributive property (also known as the FOIL method), we multiply each term in the first binomial by each term in the second binomial:

(5s + 6)(5s + 6) = (5s)(5s) + (5s)(6) + (6)(5s) + (6)(6)

Now, let's perform the multiplications:

(5s)(5s) = 25s^2 (5s)(6) = 30s (6)(5s) = 30s (6)(6) = 36

Combining these terms, we get:

25s^2 + 30s + 30s + 36

Next, we simplify by combining like terms. In this case, we have two terms with 's':

30s + 30s = 60s

So, the expanded form of (5s + 6)^2 is:

25s^2 + 60s + 36

Now that we have expanded the integrand, we can rewrite the original integral as:

∫(25s^2 + 60s + 36) ds

This expanded form is a polynomial, which is much easier to integrate term by term using the power rule and the constant multiple rule. This expansion step is crucial because it transforms a complex expression into a sum of simpler terms, each of which can be integrated directly. The ability to manipulate and simplify integrands is a key skill in calculus, and this example clearly demonstrates its importance. Mastering algebraic techniques like expanding squares is essential for successful integration.

Applying the Power Rule and Constant Multiple Rule

Having expanded the integrand, we now proceed to integrate the polynomial term by term. This involves applying two fundamental rules of integration: the power rule and the constant multiple rule. The power rule of integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is any real number except -1, and C is the constant of integration. The constant multiple rule states that ∫cf(x) dx = c∫f(x) dx, where c is a constant. These rules allow us to break down the integral into manageable parts.

We have the integral:

∫(25s^2 + 60s + 36) ds

We can split this integral into three separate integrals using the sum rule of integration, which states that the integral of a sum is the sum of the integrals:

∫(25s^2 + 60s + 36) ds = ∫25s^2 ds + ∫60s ds + ∫36 ds

Now, we apply the constant multiple rule to each integral, moving the constant coefficients outside the integral sign:

∫25s^2 ds + ∫60s ds + ∫36 ds = 25∫s^2 ds + 60∫s ds + 36∫ds

Next, we apply the power rule to each integral. For the first integral, ∫s^2 ds, n = 2, so:

∫s^2 ds = (s^(2+1))/(2+1) + C_1 = (s^3)/3 + C_1

For the second integral, ∫s ds, s is equivalent to s^1, so n = 1:

∫s ds = ∫s^1 ds = (s^(1+1))/(1+1) + C_2 = (s^2)/2 + C_2

For the third integral, ∫ds, we can think of this as ∫1 ds, which is the same as ∫s^0 ds, so n = 0:

∫ds = ∫s^0 ds = (s^(0+1))/(0+1) + C_3 = s + C_3

Now we substitute these results back into our expression:

25∫s^2 ds + 60∫s ds + 36∫ds = 25((s^3)/3) + 60((s^2)/2) + 36(s) + C

Here, we combine the constants of integration C_1, C_2, and C_3 into a single constant C. This is a common practice in indefinite integration because the constant of integration represents an arbitrary constant. By applying the power rule and the constant multiple rule systematically, we have successfully integrated each term of the polynomial. This step-by-step approach ensures accuracy and clarity in the integration process. Careful application of these rules is crucial for obtaining the correct indefinite integral.

Simplifying the Result

After applying the power rule and constant multiple rule, we obtained the expression:

25((s^3)/3) + 60((s^2)/2) + 36(s) + C

Now, we need to simplify this expression to get the final form of the indefinite integral. Simplifying the result involves performing the multiplications and combining any like terms.

First, let's perform the multiplications:

25((s^3)/3) = (25s^3)/3 60((s^2)/2) = 30s^2 36(s) = 36s

So, our expression becomes:

(25s^3)/3 + 30s^2 + 36s + C

Now, we check for any like terms that can be combined. In this case, there are no like terms, as each term has a different power of s. Therefore, the simplified form of the indefinite integral is:

(25s^3)/3 + 30s^2 + 36s + C

This is the final result of the indefinite integral of (5s + 6)^2 with respect to s. The constant of integration, C, is included to represent the family of functions that have the same derivative. The simplified expression is a polynomial, which is a common form for indefinite integrals of polynomial functions.

The simplification step is essential because it presents the result in its most concise and understandable form. It also makes it easier to use the result in further calculations or applications. By ensuring that the expression is fully simplified, we minimize the risk of errors in subsequent steps. The ability to simplify expressions is a fundamental skill in mathematics, and it is particularly important in calculus when dealing with integrals and derivatives.

Conclusion

In this article, we have successfully determined the indefinite integral of (5s + 6)^2. We began by expanding the integrand, which allowed us to rewrite the integral in a more manageable form. Then, we applied the power rule and the constant multiple rule to integrate each term of the polynomial. Finally, we simplified the result to obtain the final form of the indefinite integral: (25s^3)/3 + 30s^2 + 36s + C.

Throughout this process, we have highlighted the importance of several key concepts and techniques in calculus. Expanding the integrand is a crucial step in simplifying complex integrals. The power rule and the constant multiple rule are fundamental tools for indefinite integration. Simplifying the result ensures that the answer is in its most concise and usable form. Understanding and mastering these concepts is essential for success in calculus and related fields.

Indefinite integration is a powerful tool with numerous applications in mathematics, physics, engineering, and other disciplines. It allows us to find functions that satisfy certain derivative conditions, which is essential for solving differential equations, calculating areas and volumes, and modeling various physical phenomena. By working through this example, you have gained valuable experience in applying the techniques of indefinite integration. Practice with various examples is key to developing proficiency in this area.

The ability to determine indefinite integrals is a cornerstone of calculus, and the techniques discussed in this article provide a solid foundation for tackling more complex integration problems. We encourage you to continue practicing and exploring different types of integrals to further enhance your skills. The constant of integration, C, should always be included in indefinite integrals to represent the family of antiderivatives. Consistent practice and attention to detail will help you master the art of indefinite integration.