Indefinite Integral Of 6/x + 9/(8√x): A Step-by-Step Guide
Hey guys! Let's dive into the world of calculus and tackle a fun problem: finding the indefinite integral of the function f(x) = 6/x + 9/(8√x) on the domain of positive real numbers. If you're just starting out with integrals or need a refresher, you've come to the right place. We'll break it down step-by-step, so it's super easy to follow. So grab your pencils, and let's get started!
Understanding the Indefinite Integral
Before we jump into the problem, let's quickly recap what an indefinite integral actually is. At its core, finding the indefinite integral is the reverse process of differentiation. If you have a function, say f(x), its indefinite integral, denoted as ∫f(x) dx, gives you a family of functions whose derivative is f(x). Don't forget that sneaky little '+ C' at the end – that's the constant of integration, representing the infinite possibilities of constant terms that could disappear during differentiation.
Indefinite integrals are fundamental in calculus, and understanding them unlocks the ability to solve a wide range of problems. They're used extensively in physics, engineering, economics, and pretty much any field that involves modeling change and accumulation. Think about calculating the area under a curve, determining displacement from velocity, or even predicting population growth – integrals are your go-to tool. Mastering integration techniques isn't just about passing your calculus exam; it's about building a powerful problem-solving skill set applicable in countless real-world scenarios.
The key thing to remember is that integration is, in a way, the opposite of differentiation. If you're comfortable with derivatives, then integrals will start to feel like a natural extension of that knowledge. The more you practice, the more intuitive it becomes. Think of it like learning a new language – at first, the rules and symbols seem foreign, but with time and effort, they become second nature. So, let's embrace the challenge and dive into our specific problem!
Breaking Down the Function
Our function is f(x) = 6/x + 9/(8√x). To make integration easier, let's rewrite it slightly. Remember that √x is the same as x^(1/2), so we can rewrite the second term as 9/(8x^(1/2)). Now, let's bring that x^(1/2) up to the numerator, changing the exponent's sign. Our function now looks like this: f(x) = 6/x + (9/8)x^(-1/2). This form is much easier to work with when integrating.
When dealing with complex functions, it's often helpful to break them down into simpler terms. This makes the integration process more manageable and less intimidating. In our case, we've separated the function into two distinct parts: 6/x and (9/8)x^(-1/2). Each of these can be integrated using basic rules and techniques. Think of it like tackling a big project – you wouldn't try to do everything at once. Instead, you'd break it down into smaller, more manageable tasks. Similarly, breaking down the function simplifies the overall problem.
Recognizing these simpler components is a crucial skill in calculus. It allows you to apply specific integration rules to different parts of the function, rather than trying to find one magical formula that works for the entire expression. This approach not only makes the math easier but also enhances your understanding of how different functions behave under integration. So, remember, whenever you encounter a complex function, your first step should be to see if you can simplify it into smaller, more familiar parts. This will save you time, reduce errors, and make the whole process much smoother.
Integrating the First Term: 6/x
The first term we'll tackle is 6/x. This is a classic integral that you'll see quite often. The integral of 1/x is the natural logarithm of the absolute value of x, denoted as ln|x|. Since we have a constant multiple of 6, we simply multiply the integral by 6. So, the integral of 6/x is 6ln|x|. Remember, the absolute value is important because the natural logarithm is only defined for positive values. Since our domain is positive real numbers, we can drop the absolute value signs for now and write 6ln(x).
The integral of 1/x is a fundamental result in calculus, and it's essential to memorize it. It's one of those building blocks that you'll use again and again. The fact that it results in the natural logarithm function highlights the deep connection between exponential and logarithmic functions in calculus. When you differentiate ln(x), you get 1/x, and when you integrate 1/x, you get ln|x| + C. This inverse relationship is a cornerstone of many integration techniques.
Adding a constant multiple, like our 6, simply scales the integral. The rule here is straightforward: if you have a constant multiplying a function inside the integral, you can pull the constant outside the integral sign. This simplifies the calculation and allows you to focus on integrating the function itself. So, in our case, the integral of 6/x becomes 6 times the integral of 1/x, making it 6ln|x| + C. Mastering these basic rules and recognizing common integrals like 1/x will greatly speed up your integration process and improve your overall understanding of calculus.
Integrating the Second Term: (9/8)x^(-1/2)
Now let's move on to the second term: (9/8)x^(-1/2). For this, we'll use the power rule for integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1), provided n is not equal to -1. In our case, n = -1/2. So, we add 1 to the exponent, getting -1/2 + 1 = 1/2. Then, we divide by the new exponent, 1/2. This gives us (x^(1/2))/(1/2), which simplifies to 2x^(1/2). Don't forget the constant multiple of 9/8. So, we multiply (9/8) by 2x^(1/2), which simplifies to (9/4)x^(1/2).
The power rule of integration is another crucial tool in your calculus arsenal. It's applicable to a wide range of functions and is relatively straightforward to apply. The key is to remember the steps: add 1 to the exponent, and then divide by the new exponent. This rule works for any power of x, except for x^(-1), which, as we saw earlier, integrates to the natural logarithm. Understanding the power rule and its limitations is essential for tackling more complex integrals.
In our case, we're dealing with x^(-1/2), which is a fraction. This might seem intimidating at first, but the power rule applies just the same. Adding 1 to -1/2 gives us 1/2, and dividing by 1/2 is the same as multiplying by 2. This is where simplifying the expression becomes important. We started with (9/8)x^(-1/2), applied the power rule, and ended up with (9/4)x^(1/2). Breaking down the steps and carefully applying the rule ensures that you arrive at the correct answer. So, embrace the power rule, practice it, and you'll find it becomes a reliable friend in your integration journey.
Combining the Results and Adding the Constant of Integration
We've found the integrals of both terms: 6ln(x) for the first term and (9/4)x^(1/2) for the second term. Now, we simply add them together. This gives us 6ln(x) + (9/4)x^(1/2). But we're not quite done yet! We need to add the constant of integration, C. So, our final indefinite integral is 6ln(x) + (9/4)x^(1/2) + C.
The constant of integration, C, is a crucial part of any indefinite integral. It represents the infinite family of functions that have the same derivative. When you differentiate a function, any constant term disappears, so when you integrate, you need to account for the possibility that there was a constant term in the original function. This is why we always add '+ C' at the end of an indefinite integral.
Think of it like this: if you know the slope of a line at every point (which is what the derivative gives you), you can reconstruct the shape of the line, but you don't know its exact vertical position. The constant of integration accounts for this vertical shift. In many real-world applications, this constant can be determined using additional information, such as an initial condition. But in general, it's important to remember to include it to represent the most general form of the integral. So, our final answer, 6ln(x) + (9/4)x^(1/2) + C, is the complete and accurate indefinite integral of the given function.
Final Answer
Therefore, the indefinite integral of f(x) = 6/x + 9/(8√x) on the domain of positive real numbers is 6ln(x) + (9/4)√x + C. And there you have it! We've successfully navigated the world of indefinite integrals and found the solution to our problem. Remember, practice makes perfect, so keep tackling those integrals, and you'll become a pro in no time!
