Solving The Integral Of (6 + √x + X) / X Dx
Hey guys! Today, we're diving deep into the fascinating world of integral calculus. We're going to tackle an indefinite integral problem that might seem a bit daunting at first glance, but don't worry, we'll break it down step by step. Our mission, should we choose to accept it (and we do!), is to find the general indefinite integral of the function (6 + √x + x) / x. And remember, we can't forget our constant of integration and those absolute values where needed – they're the secret ingredients that make our solution complete!
Understanding Indefinite Integrals
Before we jump into the nitty-gritty of our specific problem, let's take a moment to refresh our understanding of indefinite integrals. Think of integration as the reverse process of differentiation. If differentiation is like taking something apart, integration is like putting it back together. An indefinite integral, specifically, is the family of all functions that have the same derivative. This is why we always add that constant of integration, often denoted as 'C,' because the derivative of a constant is always zero, meaning there are infinitely many functions that could have led to our original integrand.
In essence, finding the indefinite integral of a function f(x) means finding a function F(x) such that F'(x) = f(x). This function F(x) is called the antiderivative of f(x). Because the derivative of a constant is zero, we add '+ C' to the antiderivative to represent all possible constant terms. This '+ C' is crucial because it acknowledges that there are infinitely many functions that differ only by a constant term, all having the same derivative.
When we encounter a function like (6 + √x + x) / x, our goal is to find a function whose derivative equals this expression. This can seem like a daunting task, especially when dealing with fractions and square roots. But fear not! We have tools and techniques at our disposal. The key often lies in simplifying the integrand, breaking it down into smaller, more manageable pieces. This is where our algebraic skills come into play. By manipulating the expression, we can often rewrite it in a form that's easier to integrate using standard rules and formulas.
The power rule, the constant multiple rule, and the sum/difference rule are our best friends in this endeavor. The power rule, in particular, is a cornerstone of integration, stating that the integral of x^n is (x^(n+1))/(n+1) + C, provided n ≠ -1. The constant multiple rule allows us to pull constants out of the integral, simplifying the process. And the sum/difference rule tells us that the integral of a sum or difference is the sum or difference of the integrals. These rules, combined with a bit of algebraic manipulation, can transform complex integrals into solvable problems.
Breaking Down the Integral: A Step-by-Step Approach
Okay, let's get our hands dirty and dive into the problem at hand. We have the integral: ∫ (6 + √x + x) / x dx. The first thing we want to do is simplify the integrand. We can do this by dividing each term in the numerator by x:
∫ (6 + √x + x) / x dx = ∫ (6/x + √x/x + x/x) dx
Now, let's simplify each term individually. Remember that √x is the same as x^(1/2), so we can rewrite the expression as:
∫ (6/x + x^(1/2)/x + x/x) dx = ∫ (6/x + x^(1/2) * x^(-1) + 1) dx
Using the rule of exponents that says x^a / x^b = x^(a-b), we simplify x^(1/2) * x^(-1):
x^(1/2) * x^(-1) = x^(1/2 - 1) = x^(-1/2)
So, our integral now looks like this:
∫ (6/x + x^(-1/2) + 1) dx
Awesome! We've successfully simplified the integrand into a sum of terms that are much easier to integrate. Now, we can use the sum rule of integration, which allows us to integrate each term separately:
∫ (6/x + x^(-1/2) + 1) dx = ∫ 6/x dx + ∫ x^(-1/2) dx + ∫ 1 dx
Integrating Each Term: Applying the Rules
Let's tackle each of these integrals one by one. First up, we have ∫ 6/x dx. We can use the constant multiple rule to pull the 6 out of the integral:
∫ 6/x dx = 6 ∫ 1/x dx
Now, we need to remember a crucial integration rule: the integral of 1/x is ln|x| (the natural logarithm of the absolute value of x). The absolute value is essential here because the natural logarithm is only defined for positive values. So, we have:
6 ∫ 1/x dx = 6 ln|x| + C1
Notice that we've added a constant of integration, C1, for this part. We'll combine all the constants into a single constant at the end.
Next, we have ∫ x^(-1/2) dx. This is where the power rule comes into play. Remember, the power rule states that ∫ x^n dx = (x^(n+1))/(n+1) + C, as long as n ≠ -1. In our case, n = -1/2. So, applying the power rule, we get:
∫ x^(-1/2) dx = x^(-1/2 + 1) / (-1/2 + 1) + C2 = x^(1/2) / (1/2) + C2 = 2x^(1/2) + C2
We've again added a constant of integration, C2. Remember, x^(1/2) is the same as √x, so we can rewrite this as 2√x + C2.
Finally, we have ∫ 1 dx. This is the simplest of the three. The integral of 1 with respect to x is simply x:
∫ 1 dx = x + C3
And, you guessed it, we've added another constant of integration, C3.
Putting It All Together: The General Indefinite Integral
We've integrated each term separately, and now it's time to combine our results. We have:
∫ (6/x + x^(-1/2) + 1) dx = 6 ln|x| + C1 + 2√x + C2 + x + C3
Now, let's gather all those constants of integration (C1, C2, and C3) and combine them into a single constant, which we'll call C. Remember, the sum of any constants is still a constant, so this is perfectly valid.
6 ln|x| + C1 + 2√x + C2 + x + C3 = 6 ln|x| + 2√x + x + C
And there you have it, guys! We've found the general indefinite integral of (6 + √x + x) / x:
∫ (6 + √x + x) / x dx = 6 ln|x| + 2√x + x + C
We've successfully navigated the world of indefinite integrals, simplified a seemingly complex expression, applied the rules of integration, and remembered our constant of integration and absolute values. Give yourselves a pat on the back – you've earned it!
Key Takeaways
- Simplification is Key: Before attempting to integrate, always try to simplify the integrand using algebraic manipulations. This can make the integral much easier to solve.
- Master the Rules: Knowing the fundamental rules of integration, such as the power rule, the constant multiple rule, and the sum/difference rule, is essential.
- Remember the Constant of Integration: Don't forget to add '+ C' to every indefinite integral. It's a crucial part of the solution.
- Absolute Values Matter: When integrating 1/x, remember to use ln|x| to account for the domain of the natural logarithm.
Practice Makes Perfect
The best way to master integration is through practice. So, grab some more integral problems, put on your thinking caps, and keep those antiderivatives flowing! You've got this!
Let's embark on a journey to solve the indefinite integral of (6 + √x + x) / x dx. This problem, while seemingly intricate, can be tackled with a systematic approach and a firm understanding of integral calculus principles. Our goal is to find a function whose derivative is (6 + √x + x) / x, remembering to include the constant of integration and absolute values where necessary. This exploration will not only provide a solution but also reinforce key concepts in integration.
The Foundation: Understanding Indefinite Integrals and Their Properties
At its core, integration is the reverse process of differentiation. Imagine differentiation as dissecting a complex structure into its fundamental components; integration, then, is the art of reconstructing the original structure from these components. An indefinite integral represents the family of all functions sharing the same derivative. This is why the constant of integration, denoted as 'C', is paramount. It acknowledges the infinite possibilities, as the derivative of any constant is zero, meaning numerous functions differing only by a constant term could yield the same derivative.
The indefinite integral of a function f(x) is a function F(x) such that F'(x) = f(x). F(x) is known as the antiderivative of f(x). The '+ C' in F(x) + C signifies the constant of integration, accommodating the myriad constant terms that could exist. For a function like (6 + √x + x) / x, the challenge lies in finding a function whose derivative matches this expression. This can initially seem daunting, particularly with fractions and square roots involved. However, the key is to simplify the integrand, dissecting it into manageable parts. This is where algebraic prowess becomes invaluable. By manipulating the expression, we can often recast it into a form more amenable to integration using established rules and formulas.
The power rule, the constant multiple rule, and the sum/difference rule are our trusty allies in this endeavor. The power rule, a cornerstone of integration, states that the integral of x^n is (x^(n+1))/(n+1) + C, provided n ≠ -1. The constant multiple rule allows us to extract constants from the integral, simplifying the process. The sum/difference rule dictates that the integral of a sum or difference is the sum or difference of the integrals. These rules, when combined with strategic algebraic manipulation, can transform seemingly complex integrals into solvable puzzles.
Step-by-Step Solution: Unraveling the Integral of (6 + √x + x) / x
Now, let's roll up our sleeves and tackle the specific integral at hand: ∫ (6 + √x + x) / x dx. The initial step is to simplify the integrand. We can achieve this by dividing each term in the numerator by x:
∫ (6 + √x + x) / x dx = ∫ (6/x + √x/x + x/x) dx
Next, let's simplify each term individually. Recognizing that √x is equivalent to x^(1/2), we can rewrite the expression as:
∫ (6/x + x^(1/2)/x + x/x) dx = ∫ (6/x + x^(1/2) * x^(-1) + 1) dx
Applying the exponent rule x^a / x^b = x^(a-b), we simplify x^(1/2) * x^(-1):
x^(1/2) * x^(-1) = x^(1/2 - 1) = x^(-1/2)
Thus, our integral now takes a more manageable form:
∫ (6/x + x^(-1/2) + 1) dx
Excellent! We've successfully streamlined the integrand into a sum of terms that are far easier to integrate. Now, we can invoke the sum rule of integration, which permits us to integrate each term independently:
∫ (6/x + x^(-1/2) + 1) dx = ∫ 6/x dx + ∫ x^(-1/2) dx + ∫ 1 dx
Integrating Term by Term: Applying the Integral Calculus Arsenal
Let's methodically integrate each term. First, we address ∫ 6/x dx. Employing the constant multiple rule, we can extract the 6 from the integral:
∫ 6/x dx = 6 ∫ 1/x dx
Now, we recall a fundamental integration rule: the integral of 1/x is ln|x| (the natural logarithm of the absolute value of x). The absolute value is crucial because the natural logarithm is defined only for positive values. Therefore:
6 ∫ 1/x dx = 6 ln|x| + C1
Note the addition of the constant of integration, C1, for this segment. We will consolidate all constants into a single constant at the culmination of the process.
Next, we tackle ∫ x^(-1/2) dx. This is where the power rule shines. The power rule stipulates that ∫ x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1. In this instance, n = -1/2. Applying the power rule yields:
∫ x^(-1/2) dx = x^(-1/2 + 1) / (-1/2 + 1) + C2 = x^(1/2) / (1/2) + C2 = 2x^(1/2) + C2
Again, we incorporate the constant of integration, C2. Recognizing that x^(1/2) is synonymous with √x, we can rewrite this as 2√x + C2.
Finally, we address ∫ 1 dx. This is the simplest of the three. The integral of 1 with respect to x is simply x:
∫ 1 dx = x + C3
And, as anticipated, we've added yet another constant of integration, C3.
The Grand Finale: Constructing the General Indefinite Integral
Having integrated each term individually, we now assemble our results. We have:
∫ (6/x + x^(-1/2) + 1) dx = 6 ln|x| + C1 + 2√x + C2 + x + C3
Now, we consolidate the constants of integration (C1, C2, and C3) into a single constant, denoted as C. Since the sum of constants is invariably a constant, this operation is mathematically sound.
6 ln|x| + C1 + 2√x + C2 + x + C3 = 6 ln|x| + 2√x + x + C
Thus, we've successfully determined the general indefinite integral of (6 + √x + x) / x:
∫ (6 + √x + x) / x dx = 6 ln|x| + 2√x + x + C
We've navigated the intricacies of indefinite integrals, simplified a complex expression, applied the rules of integration, and conscientiously included the constant of integration and absolute values. This is a testament to a methodical approach and a solid grasp of integral calculus principles.
Key Principles and Takeaways
- Strategic Simplification: Before embarking on integration, always strive to simplify the integrand using algebraic manipulations. This can significantly ease the integration process.
- Mastery of Integration Rules: A thorough understanding of fundamental integration rules, such as the power rule, the constant multiple rule, and the sum/difference rule, is indispensable.
- The Imperative Constant of Integration: Never omit the '+ C' in an indefinite integral. It's a non-negotiable element of a complete solution.
- The Significance of Absolute Values: When integrating 1/x, always employ ln|x| to accommodate the domain of the natural logarithm.
The Path to Proficiency: Practice and Perseverance
The journey to mastering integration is paved with practice. Seek out diverse integral problems, don your analytical hat, and keep those antiderivatives flowing. With consistent effort and a solid understanding of the principles, you'll undoubtedly conquer the realm of integral calculus.
Let's tackle the challenge of finding the indefinite integral of (6 + √x + x) / x dx. This problem provides a great opportunity to reinforce our understanding of integration techniques, including simplification, application of integral rules, and the crucial inclusion of the constant of integration. We'll break down the process step-by-step, ensuring clarity and comprehension.
Understanding the Basics: Indefinite Integrals and Their Properties
At its core, an indefinite integral represents the family of all functions that share the same derivative. It's the inverse operation of differentiation. Think of it as piecing together a puzzle, where the derivative is the disassembled pieces, and the integral is the complete picture. The constant of integration, denoted as 'C,' plays a vital role in this concept. Since the derivative of a constant is always zero, there are infinitely many functions that could have the same derivative, differing only by a constant value. Therefore, '+ C' accounts for all possible constant terms in the antiderivative.
The goal when finding the indefinite integral of a function f(x) is to determine a function F(x) such that F'(x) = f(x). This F(x) is known as the antiderivative. For our specific problem, (6 + √x + x) / x, we need to find a function whose derivative matches this expression. This may seem complex initially, but with strategic simplification and the application of integral rules, it becomes manageable. The key is to break down the integrand into smaller, more easily integrable components.
Several fundamental rules of integration will be our guides in this process. The power rule is a cornerstone, stating that ∫ x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1. The constant multiple rule allows us to factor out constants from the integral, simplifying the calculation. The sum/difference rule states that the integral of a sum or difference of terms is equal to the sum or difference of their individual integrals. These rules, combined with algebraic manipulation, are our tools for solving a wide range of integral problems.
Step-by-Step Solution: Integrating (6 + √x + x) / x
Let's dive into the problem: ∫ (6 + √x + x) / x dx. The first and often most crucial step is to simplify the integrand. We can do this by dividing each term in the numerator by x:
∫ (6 + √x + x) / x dx = ∫ (6/x + √x/x + x/x) dx
Now, let's simplify each resulting term individually. Recall that √x is equivalent to x^(1/2). We can rewrite the expression as:
∫ (6/x + x^(1/2)/x + x/x) dx = ∫ (6/x + x^(1/2) * x^(-1) + 1) dx
Using the rule of exponents that states x^a / x^b = x^(a-b), we simplify x^(1/2) * x^(-1):
x^(1/2) * x^(-1) = x^(1/2 - 1) = x^(-1/2)
This gives us a simplified integral:
∫ (6/x + x^(-1/2) + 1) dx
We've successfully transformed the integrand into a sum of more manageable terms. Next, we apply the sum rule of integration, which allows us to integrate each term separately:
∫ (6/x + x^(-1/2) + 1) dx = ∫ 6/x dx + ∫ x^(-1/2) dx + ∫ 1 dx
Term-by-Term Integration: Applying the Integral Rules
Let's integrate each term one by one. First, consider ∫ 6/x dx. We can use the constant multiple rule to move the 6 outside the integral:
∫ 6/x dx = 6 ∫ 1/x dx
Now, we recall a fundamental integration rule: the integral of 1/x is ln|x| (the natural logarithm of the absolute value of x). The absolute value is essential here because the natural logarithm is only defined for positive values. So:
6 ∫ 1/x dx = 6 ln|x| + C1
We've added a constant of integration, C1, for this term. We'll combine all the constants into a single constant at the end.
Next, we have ∫ x^(-1/2) dx. This is where the power rule comes into play. Recall that the power rule states ∫ x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1. In our case, n = -1/2. Applying the power rule gives us:
∫ x^(-1/2) dx = x^(-1/2 + 1) / (-1/2 + 1) + C2 = x^(1/2) / (1/2) + C2 = 2x^(1/2) + C2
We add another constant of integration, C2. Since x^(1/2) is the same as √x, we can rewrite this as 2√x + C2.
Finally, we have ∫ 1 dx. This is the simplest of the three. The integral of 1 with respect to x is simply x:
∫ 1 dx = x + C3
And, of course, we add another constant of integration, C3.
Combining the Results: The General Indefinite Integral
Now that we've integrated each term, we combine our results:
∫ (6/x + x^(-1/2) + 1) dx = 6 ln|x| + C1 + 2√x + C2 + x + C3
We gather all the constants of integration (C1, C2, and C3) and combine them into a single constant, which we'll call C. The sum of any constants is still a constant, so this is valid.
6 ln|x| + C1 + 2√x + C2 + x + C3 = 6 ln|x| + 2√x + x + C
Thus, the general indefinite integral of (6 + √x + x) / x is:
∫ (6 + √x + x) / x dx = 6 ln|x| + 2√x + x + C
We've successfully navigated the process of indefinite integration, simplified the integrand, applied the appropriate rules, and remembered the crucial constant of integration and absolute values. This demonstrates a strong understanding of integral calculus principles.
Key Concepts and Takeaways
- Simplify First: Always attempt to simplify the integrand before integrating. This often makes the process much easier.
- Master the Integral Rules: A solid understanding of fundamental integral rules, such as the power rule, constant multiple rule, and sum/difference rule, is essential.
- Don't Forget the Constant of Integration: Always add '+ C' to indefinite integrals. It's a critical part of the solution.
- Absolute Values for Logarithms: Remember to use ln|x| when integrating 1/x, as the natural logarithm is only defined for positive values.
Practice for Proficiency
The key to mastering integration is consistent practice. Work through a variety of integral problems, and don't hesitate to review the rules and techniques as needed. With persistence and a solid foundation, you'll become proficient in solving indefinite integrals.