Indefinite Integral Of (12/√x + 12√x) Dx: A Step-by-Step Guide
Have you ever stumbled upon an integral that looks intimidating at first glance? Integrals like ∫(12/√x + 12√x) dx can seem complex, but with a systematic approach, they become manageable. In this comprehensive guide, we will walk you through the step-by-step process of evaluating this indefinite integral. Whether you're a student tackling calculus or just brushing up on your math skills, this explanation is designed to help you understand each stage clearly and effectively. Let’s break down this problem together and turn it into a piece of cake!
Understanding the Basics of Indefinite Integrals
Before we dive into the specific problem, let's refresh our understanding of indefinite integrals. Indefinite integrals, at their core, are about finding the antiderivative of a function. Think of it as reversing the process of differentiation. When you differentiate a function, you find its rate of change. Integration, in a way, is about finding the original function given its rate of change. The result of an indefinite integral is not a single function, but rather a family of functions that differ by a constant. This is why we always add the constant of integration, often denoted as 'C', to the final result. So, when you see ∫f(x) dx, you're essentially asking: "What function, when differentiated, gives me f(x)?"
Now, why is this important? Indefinite integrals are fundamental in many areas of mathematics, physics, and engineering. They are used to calculate areas under curves, solve differential equations, and model various physical phenomena. Mastering indefinite integrals is crucial for anyone pursuing studies or careers in these fields. The key is to break down the integral into simpler, manageable parts, and apply the basic rules of integration. Remember, practice makes perfect, and the more you work with integrals, the more comfortable you’ll become with them. Let's get started by looking at the specific techniques we'll use for our problem: ∫(12/√x + 12√x) dx.
Breaking Down the Integral
Our task is to find the indefinite integral of ∫(12/√x + 12√x) dx. The first step in tackling this integral is to break it down into simpler parts. This is a common strategy in calculus because it allows us to apply integration rules more easily. The integral of a sum is equal to the sum of the integrals, a property that makes our job much easier. We can rewrite the integral as follows:
∫(12/√x + 12√x) dx = ∫(12/√x) dx + ∫(12√x) dx
Now we have two separate integrals to solve, which are much more manageable. The next step is to address each of these integrals individually. We'll start by focusing on the first integral, ∫(12/√x) dx. The key here is to recognize that 1/√x can be rewritten using exponents, which will allow us to apply the power rule of integration. Remember, mathematical problems often become simpler when we manipulate the form of the expressions involved. By breaking the integral into smaller pieces, we've already made significant progress. Each term now looks less daunting, and we can approach them one at a time, using familiar integration techniques.
Rewriting the Terms Using Exponents
Before we can directly integrate, it's essential to rewrite the terms inside the integrals using exponents. This makes it easier to apply the power rule for integration. Let's start with the first term, 12/√x. Remember that the square root of x, denoted as √x, can be written as x^(1/2). Therefore, 1/√x can be written as 1/x^(1/2). To bring this term into the numerator, we can use the property that 1/x^n = x^(-n). So, 1/x^(1/2) becomes x^(-1/2).
Now, our first integral ∫(12/√x) dx can be rewritten as ∫12x^(-1/2) dx. This form is much easier to work with because it directly fits the power rule for integration. Next, let's consider the second term, 12√x. As we mentioned earlier, √x is equivalent to x^(1/2). Therefore, the second integral ∫(12√x) dx can be rewritten as ∫12x^(1/2) dx. Now both of our integrals are in a form where we can easily apply the power rule. Rewriting terms using exponents is a crucial technique in calculus, and it's something you'll use frequently when evaluating integrals. By making this simple transformation, we've set ourselves up for the next step: applying the power rule for integration.
Applying the Power Rule for Integration
The power rule for integration is a fundamental concept that simplifies the process of finding indefinite integrals. The power rule 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. This rule is derived from the reverse process of the power rule for differentiation, where the power decreases by one. To apply the power rule to our integrals, let's revisit our rewritten integrals:
- ∫12x^(-1/2) dx
- ∫12x^(1/2) dx
For the first integral, we have n = -1/2. Applying the power rule, we get:
∫12x^(-1/2) dx = 12 * (x^(-1/2 + 1))/(-1/2 + 1) + C
Simplifying the exponent and the denominator, we have:
12 * (x^(1/2))/(1/2) + C = 12 * 2x^(1/2) + C = 24x^(1/2) + C
Now, let's apply the power rule to the second integral, ∫12x^(1/2) dx, where n = 1/2:
∫12x^(1/2) dx = 12 * (x^(1/2 + 1))/(1/2 + 1) + C
Simplifying the exponent and the denominator, we get:
12 * (x^(3/2))/(3/2) + C = 12 * (2/3)x^(3/2) + C = 8x^(3/2) + C
By applying the power rule, we've successfully integrated each term separately. The next step is to combine these results to get the complete indefinite integral. Remember, the power rule is a cornerstone of integral calculus, and understanding how to apply it correctly is essential for solving a wide range of integration problems.
Combining the Results
Now that we've applied the power rule to each integral separately, it's time to combine the results to find the overall indefinite integral. We found that:
∫12x^(-1/2) dx = 24x^(1/2) + C₁ ∫12x^(1/2) dx = 8x^(3/2) + C₂
To get the indefinite integral of the original expression, we simply add these two results together:
∫(12/√x + 12√x) dx = 24x^(1/2) + 8x^(3/2) + C₁ + C₂
Since C₁ and C₂ are both arbitrary constants, we can combine them into a single constant, C. This simplifies our expression to:
24x^(1/2) + 8x^(3/2) + C
It's often helpful to rewrite the terms back in their original form, using square roots instead of fractional exponents. Recall that x^(1/2) is the same as √x, and x^(3/2) is the same as x√x. Therefore, our integral can be written as:
24√x + 8x√x + C
This is the indefinite integral of the original expression. By combining the results of our individual integrations, we've arrived at the final solution. It’s a testament to the power of breaking down complex problems into simpler parts and then piecing the solutions back together. In the next section, we'll explore how to verify our answer and ensure its accuracy.
Verifying the Solution
After finding the indefinite integral, it's always a good practice to verify the solution. The easiest way to do this is by differentiating the result and checking if it matches the original integrand. If the derivative of our result is the same as the function we integrated, then we know our solution is correct. Our solution is:
24√x + 8x√x + C
Let's differentiate this expression with respect to x. Remember, differentiation is the inverse operation of integration, so if we've done our integration correctly, we should end up with the original function. First, rewrite the expression using exponents:
24x^(1/2) + 8x^(3/2) + C
Now, apply the power rule for differentiation, which states that d/dx(x^n) = nx^(n-1). The derivative of a constant C is 0. Differentiating each term, we get:
d/dx(24x^(1/2)) = 24 * (1/2)x^(1/2 - 1) = 12x^(-1/2) d/dx(8x^(3/2)) = 8 * (3/2)x^(3/2 - 1) = 12x^(1/2) d/dx(C) = 0
Adding these derivatives together, we have:
12x^(-1/2) + 12x^(1/2)
Now, rewrite the terms using radicals:
12/√x + 12√x
This is exactly the original integrand, which means our indefinite integral is correct! Verifying the solution is a critical step in any calculus problem. It gives you confidence in your answer and helps catch any potential errors. By differentiating our result and confirming that it matches the original function, we've ensured the accuracy of our solution.
Conclusion
In conclusion, we have successfully evaluated the indefinite integral ∫(12/√x + 12√x) dx. By breaking down the integral into simpler parts, rewriting terms using exponents, applying the power rule for integration, combining the results, and verifying the solution, we have demonstrated a systematic approach to solving this type of problem. Remember, the key to mastering calculus is practice and a clear understanding of fundamental concepts.
This step-by-step guide has shown how to tackle an integral that might initially seem daunting. Each step, from breaking down the integral to verifying the solution, is crucial in ensuring accuracy and understanding. Integrals like these are common in various fields, including physics and engineering, so mastering them is a valuable skill. We encourage you to continue practicing and applying these techniques to other problems. With time and effort, you’ll become more confident and proficient in solving indefinite integrals.
For further learning and practice on integration techniques, you might find resources on websites like Khan Academy's Integral Calculus incredibly helpful.
