Solving The Integral Of Ln(x) / X * √(1 + (ln(x))^2) - Step-by-Step

Hey math enthusiasts! Today, we're diving into a fascinating integral: the integral of ln(x) / x * √(1 + (ln(x))^2) dx. Don't worry if it looks a bit intimidating at first – we'll break it down step by step to make it super clear and manageable. This integral is a great example of how a clever substitution can simplify a seemingly complex expression, and it's a valuable technique to add to your problem-solving toolkit. So, grab your pencils, open your notebooks, and let's get started!

Understanding the Problem: The Integral Unveiled

First off, let's take a closer look at the integral itself: ∫ (ln(x) / x) * √(1 + (ln(x))^2) dx. At first glance, this integral might seem like a handful, with the natural logarithm ln(x), its derivative 1/x, and a square root all in the mix. But don't be discouraged! The key to tackling this problem is recognizing the relationship between the different parts of the integrand, specifically the natural logarithm ln(x) and its derivative, which is 1/x. This relationship is a prime indicator that we can use a substitution to simplify the expression and make it easier to solve. The presence of the square root of a function of ln(x) also hints that a clever substitution could greatly simplify the integral. Our goal is to transform this complex integral into a more manageable form, allowing us to apply the fundamental rules of integration. The presence of ln(x) and its derivative 1/x is a telltale sign that we should look to substitute ln(x) with another variable, which will help us solve the integral more easily. So, let's gear up to use the substitution method.

Now, a little bit about why this integral is interesting: It showcases a common technique in calculus, which is the skillful use of substitutions to make integrals easier to solve. It's a fantastic example that helps build a solid foundation in integral calculus. Mastering this method will not only help you solve this specific problem but also help you solve a whole variety of other problems in your math classes. Understanding the steps involved in solving such an integral provides valuable insights into how to approach other similar problems you might come across in calculus. This is a crucial area in mathematics because it gives you the tools needed to figure out areas, volumes, and many other mathematical problems. Remember, the goal is to transform the integral into a simpler form that we can solve using fundamental integration rules. The relationship between ln(x) and 1/x is critical, so keep a keen eye out for similar patterns when approaching other integrals. Always remember that practice makes perfect, so be sure to try other examples to perfect your skills and understanding of integration techniques. The journey through calculus can be tricky, but knowing how to use different integration methods, such as substitution, will help you overcome any challenge you encounter. The integral is a fine example of how calculus concepts are actually used in practice, showing that mathematics is not just theory but a useful tool. This integral emphasizes the importance of substitution as a vital technique to simplify complex integrations.

Step-by-Step Solution: Cracking the Integral

Alright, guys, let's get down to business and walk through the steps of solving this integral together. Remember, the substitution method is your friend here!

Step 1: Choosing the Right Substitution

First things first: We need to choose the right substitution. Looking at the integral, it's clear that the expression ln(x) and its derivative 1/x are closely related. This means substituting u = ln(x) is our best bet. This substitution aims to simplify the expression under the square root, making the integral more manageable. This is because the derivative of ln(x) appears in the integral, which allows us to cancel out terms and simplify the expression. The goal is to make the integral look simpler and be easier to solve. This is a common strategy when dealing with integrals involving natural logarithms and their derivatives. So, let's replace ln(x) with a new variable, u, to make the calculations easier. A well-chosen substitution can significantly reduce the complexity of the integral, helping us to apply basic integration rules. Make sure to carefully consider the entire expression and choose the substitution that will simplify things most efficiently. Remember, the main goal is to transform the integral into a more manageable form, enabling us to use the standard rules of integration.

When choosing your substitution, always keep in mind the goal of simplifying the integral. The correct substitution will help us to eliminate or greatly reduce the complexity of our expression. The choice of u = ln(x) will greatly help simplify the expression and make it more manageable. With u = ln(x), we will be able to easily find the derivative, which will simplify the integral even further, leading to an easy solution. The key to mastering integration is the ability to recognize patterns and choose the substitution that makes things the easiest. Always choose the substitution that best simplifies your expression, allowing you to use the standard rules of integration. You will become better at picking the right substitution through practice. Remember, the correct substitution is what will get us on the right path to successfully solving the integral.

Step 2: Differentiating and Finding du

Now that we have u = ln(x), we need to find its derivative, which will help us rewrite the integral in terms of u. Differentiating both sides with respect to x, we get du/dx = 1/x. This means du = (1/x) dx. This step is crucial because it allows us to substitute (1/x) dx with du in the integral. Recognizing that (1/x) dx is already present in our original integral is a key part of making this substitution work. The process of differentiating u = ln(x) gives us the expression we need for substitution in the original integral. The derivative du = (1/x) dx is a vital piece because it allows us to convert the integral in terms of x into an integral in terms of u. The goal here is to rewrite the integral solely in terms of u, making it simpler to solve. Always remember that the derivative is the key that unlocks the door to substitution. This step transforms the integral and prepares it for simplification, so it is a really important one. By doing this step, we ensure that the expression is in terms of the new variable and that it can be simplified. The ability to find du correctly is a crucial skill in integration.

Finding du correctly allows us to switch from the variable x to the variable u, and this is extremely important for simplifying the integral. Without du, our substitution wouldn't work. The equation du = (1/x) dx allows us to replace (1/x) dx in the integral with du. This substitution greatly simplifies our integral, making it easier to solve. Remember, du is the derivative of u with respect to x, which allows us to find the integral with respect to the new variable. The value of du shows the relation between the u and x variables in the integral. Using du is essential to make sure the integration is correct and accurate. Practice makes perfect, so work on finding du until you feel really comfortable with it. Each time you integrate, it's essential to ensure you correctly determine the corresponding du value.

Step 3: Rewriting the Integral in Terms of u

Now, let's rewrite the integral using our substitution. We have: ∫ (ln(x) / x) * √(1 + (ln(x))^2) dx. Substituting u = ln(x) and du = (1/x) dx, we get: ∫ u * √(1 + u^2) du. This looks much simpler, doesn’t it? This transformation is where the magic happens. We've transformed a complicated integral involving ln(x) and its derivative into a much more straightforward one. The simpler the integral becomes, the easier it is to solve it. After the substitution, the integral is ready to be solved using basic integration rules. Remember, it is important to rewrite the integral completely in terms of u. The goal is to make the expression simpler to solve using the basic rules of integration. This step simplifies the problem significantly, helping us to apply basic integration rules. This rewriting of the integral makes it more manageable and straightforward, leading us towards a solution. The substitution reduces the complexity of the integral, allowing us to find the solution more easily. This step highlights the power of substitution in integral calculus and greatly simplifies the expression.

By simplifying the integral, we are one step closer to solving it, and this is another example of why the substitution method is so useful. The substitution transforms a complex expression into something that is easier to integrate. The new form of the integral is much more straightforward and approachable for further steps. This transformation makes the problem much simpler and more manageable. The simpler form of the integral helps us apply the integration rules and get closer to finding our answer.

Step 4: Solving the Simplified Integral

Now we have: ∫ u * √(1 + u^2) du. To solve this, we can use another substitution, let's call it v = 1 + u^2. Then, dv/du = 2u, so dv = 2u du. We can rewrite this as u du = (1/2) dv. We are going to simplify the integral even further by using another substitution. This is also a common trick in integration. This new substitution is meant to simplify the expression even further, which will allow us to find the final result with ease. We make another substitution to simplify the integral even more. This type of double substitution is really useful for solving complicated problems. Recognizing the structure of the integrand allows us to make the new substitution which simplifies things again. Now the integral is easier to solve, we will use another variable, v, to do it. The key is to simplify the integral and then solve it. This secondary substitution transforms the integral to make it easier to solve using basic rules.

Using this new substitution, the integral becomes: ∫ √(v) * (1/2) dv, which simplifies to (1/2) ∫ v^(1/2) dv. Using the power rule for integration, we get: (1/2) * (2/3) * v^(3/2) + C = (1/3) * v^(3/2) + C. This step uses the power rule for integration to solve the integral with respect to v. The power rule for integration is now being used to find the integral with respect to v, making it easier to solve. The power rule allows us to perform the final integration, which gives us an answer in terms of v. Remember the constant of integration, C, because that's a really important thing. The final result is the integral with respect to the new variable, v. Now that the integration has been done, we're very close to finding the solution. The application of the power rule is straightforward but crucial for getting the correct result. At this stage, we are very close to finding the solution, making sure that we don't forget the integration constant C.

Step 5: Back-Substituting and Finding the Final Answer

We've solved the integral in terms of v, but the original problem was in terms of x. So, we need to substitute back to get our final answer. Remember, v = 1 + u^2 and u = ln(x). So, substituting back, we get: (1/3) * (1 + u^2)^(3/2) + C = (1/3) * (1 + (ln(x))^2)^(3/2) + C. And there you have it, folks! That's the solution to the integral! We're back in the domain of the original variable, x. We're putting it all together in this step, to express the final answer in terms of the original variable. This is a crucial step in the whole process of solving the integral. The final answer is expressed in terms of the original variable x, and now we are able to express our final result. This step puts everything back together and expresses the final result. Remember to put back the original variables so that your answer makes sense for the original problem. Now the solution is in the original form, in terms of x. The substitution step allows us to express the integral in terms of the original variables, x. We finally have the solution to the original integral in terms of x.

Conclusion: Wrapping It Up

Congratulations, guys! You've successfully navigated the integral of ln(x) / x * √(1 + (ln(x))^2) dx using the substitution method. We started with a complex-looking integral and, through careful substitutions, were able to simplify it into something we could solve using basic integration rules. Remember, practice is key. Try solving similar integrals on your own. Keep experimenting with different substitutions and see how they change the integral. The more you practice, the better you'll become at recognizing patterns and choosing the right substitutions. You will gain confidence with more practice. Use this solution as a reference for solving other similar integrals. Remember that solving complex integrals requires a series of steps and requires a good understanding of calculus. Solving such problems will provide you with a deeper understanding of calculus concepts. The goal is to build your skills and get better at solving all kinds of integration problems. Always take your time and break down the problems into small steps. Keep practicing, and you'll find yourself solving integrals with ease! This experience will give you the skills needed to tackle the most difficult problems. Keep in mind that a good understanding of calculus concepts and constant practice are essential.

So, keep up the great work, and keep exploring the wonderful world of calculus! You've totally got this! Feel free to ask any questions you have. Happy integrating!