Solving ∫ Dx / [x(1 + Ln X)] From 1 To E^3: A Step-by-Step Guide

by Alex Johnson 65 views

In this article, we'll dive deep into the process of evaluating the definite integral ∫[1 to e^3] dx / [x(1 + ln x)]. This integral, while seemingly complex at first glance, can be solved elegantly using the method of u-substitution. We'll break down each step, ensuring clarity and understanding, so you can confidently tackle similar problems in the future. Whether you're a student grappling with calculus or simply someone with a keen interest in mathematical problem-solving, this guide will provide you with the tools and insights you need.

Understanding the Integral

Before we jump into the solution, let's take a moment to understand the integral itself. We're dealing with a definite integral, meaning we're calculating the area under the curve of the function f(x) = 1 / [x(1 + ln x)] between the limits of integration x = 1 and x = e^3. The integrand, 1 / [x(1 + ln x)], involves a natural logarithm function, which often suggests that u-substitution might be a fruitful approach. Recognizing these key features is crucial for choosing the right strategy. The presence of both 'x' and 'ln x' in the denominator hints at a possible substitution that simplifies the expression. This initial assessment is a critical step in problem-solving, as it guides our choice of techniques and sets the stage for a smoother solution process. By carefully observing the structure of the integral, we can make informed decisions about the most effective method to employ.

The Power of u-Substitution

The u-substitution technique is a powerful tool in calculus that allows us to simplify integrals by replacing a complex expression with a single variable, 'u'. The key is to identify a suitable substitution that simplifies the integrand and makes it easier to integrate. In this case, a natural choice for 'u' is the expression (1 + ln x), as its derivative, 1/x, appears in the integrand. This strategic selection is the cornerstone of the method, allowing us to transform the integral into a more manageable form. The process involves not only choosing 'u' but also finding the relationship between 'du' and 'dx', which is crucial for the substitution to work effectively. By carefully selecting 'u' and manipulating the differential elements, we can often unravel seemingly intricate integrals and bring them within the reach of standard integration techniques. This technique exemplifies the beauty of calculus, where clever substitutions can transform challenging problems into straightforward exercises.

Step-by-Step Solution

Let's walk through the solution step-by-step:

  1. Choose u: Let u = 1 + ln x.
  2. Find du: Differentiating both sides with respect to x, we get du/dx = 1/x, which implies du = (1/x) dx.
  3. Change the limits of integration: Since we're dealing with a definite integral, we need to change the limits of integration to reflect the new variable 'u'.
    • When x = 1, u = 1 + ln(1) = 1 + 0 = 1.
    • When x = e^3, u = 1 + ln(e^3) = 1 + 3 = 4.
  4. Substitute: Now we substitute u and du into the integral, and we also change the limits of integration. The integral becomes: ∫[1 to 4] (1/u) du
  5. Integrate: The integral of 1/u is ln|u|. So, we have: ln|u| evaluated from 1 to 4.
  6. Evaluate: We now evaluate the antiderivative at the upper and lower limits of integration: ln(4) - ln(1) = ln(4) - 0 = ln(4).
  7. Simplify: Since 4 = 2^2, we can further simplify ln(4) as 2ln(2). Therefore, the final result is 2ln(2).

Each of these steps is crucial for a complete and accurate solution. The careful selection of 'u', the correct calculation of 'du', the transformation of the limits of integration, and the accurate evaluation of the antiderivative all contribute to the final answer. This methodical approach not only provides the solution but also reinforces a systematic problem-solving mindset, which is invaluable in mathematics and beyond.

Common Mistakes to Avoid

When tackling integrals using u-substitution, there are several common pitfalls to be aware of. One frequent mistake is forgetting to change the limits of integration when dealing with a definite integral. Failing to do so means you'll need to substitute back to the original variable 'x' after integrating, which adds an extra step and increases the chance of error. Another common error is incorrectly calculating 'du' or making a mistake during the substitution process. A careful and methodical approach is essential to avoid these slips. Additionally, students sometimes choose an inappropriate substitution for 'u', which can complicate the integral rather than simplify it. Practice and familiarity with different types of integrals are key to developing the intuition for selecting the optimal substitution. By being mindful of these common errors and taking the time to double-check your work, you can significantly improve your accuracy and confidence in solving integrals.

Alternative Approaches

While u-substitution is the most straightforward method for solving this particular integral, it's worth noting that other approaches might exist, although they may be less efficient. For instance, one could attempt to use integration by parts, but this would likely lead to a more complex expression. Recognizing the suitability of u-substitution in this case highlights the importance of developing a repertoire of integration techniques and being able to identify the most appropriate method for a given problem. Exploring alternative approaches, even if they are not the most efficient, can deepen your understanding of integration and enhance your problem-solving skills. The ability to compare and contrast different methods is a hallmark of mathematical maturity, allowing you to choose the most elegant and effective solution strategy.

Practice Makes Perfect

The key to mastering integral calculus, like any mathematical skill, is practice. Working through a variety of problems will help you develop your intuition for choosing the right integration techniques and avoid common mistakes. Start with simpler integrals and gradually progress to more challenging ones. Pay close attention to the structure of the integrand and think about which substitution might simplify the expression. Don't be afraid to make mistakes – they are a valuable part of the learning process. By consistently practicing and reviewing your work, you'll build confidence and proficiency in solving integrals. Remember, each problem you solve is a step forward in your mathematical journey.

Conclusion

In conclusion, we've successfully evaluated the definite integral ∫[1 to e^3] dx / [x(1 + ln x)] using the method of u-substitution. The result, 2ln(2), demonstrates the power and elegance of this technique. By carefully choosing our substitution, changing the limits of integration, and applying the fundamental theorem of calculus, we were able to solve a seemingly complex problem in a clear and concise manner. Remember to practice regularly and be mindful of common mistakes to continue honing your integration skills. Explore further resources and continue to challenge yourself with new problems to deepen your understanding of calculus. Happy integrating!

For further exploration of integration techniques, you might find the resources at Khan Academy's Integral Calculus section helpful.