What Is The Integral Of Xlnx

What is the Integral of xlnx? A Comprehensive Guide

The integral of xlnx is a classic example in calculus that elegantly demonstrates the power of integration by parts. While seemingly straightforward, understanding its solution requires a grasp of fundamental integration techniques and a bit of algebraic manipulation. This comprehensive guide will delve into the process step-by-step, exploring various approaches and providing ample context for a complete understanding.

Understanding Integration by Parts

Before tackling the integral of xlnx, let's review the core principle of integration by parts. This technique is invaluable when dealing with integrands that are products of two functions. The formula is derived from the product rule of differentiation and is expressed as:

∫u dv = uv - ∫v du

Where:

• ∫ represents the integral sign.
• u and v are functions of x.
• du and dv are their respective differentials.

The key to successfully applying integration by parts lies in strategically choosing u and dv. A common mnemonic device is "LIATE," which prioritizes the order of function types:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions (like polynomials)
• Trigonometric functions
• Exponential functions

This order suggests that choosing the function higher on the list as u often simplifies the integration process.

Solving the Integral of xlnx

Now, let's apply integration by parts to solve ∫xlnx dx. Following the LIATE rule, we choose:

• u = ln x (Logarithmic function)
• dv = x dx (Algebraic function)

Next, we find the derivatives and integrals:

• du = (1/x) dx (Derivative of ln x)
• v = (1/2)x² (Integral of x dx)

Substituting these values into the integration by parts formula:

∫xlnx dx = (ln x) * ((1/2)x²) - ∫((1/2)x²) * (1/x) dx

Simplifying the expression:

∫xlnx dx = (1/2)x²lnx - (1/2)∫x dx

Now, we integrate the remaining term:

∫x dx = (1/2)x²

Therefore, the final solution is:

∫xlnx dx = (1/2)x²lnx - (1/4)x² + C

Where C is the constant of integration. This constant is crucial because the derivative of a constant is zero, meaning multiple functions can have the same derivative.

Verifying the Solution Through Differentiation

To ensure the accuracy of our solution, we can differentiate the result and see if we obtain the original integrand, xlnx. Using the product rule and chain rule:

d/dx [(1/2)x²lnx - (1/4)x² + C] = (1/2)(2xlnx + x²/x) - (1/2)x

Simplifying:

= xlnx + (1/2)x - (1/2)x = xlnx

This confirms the correctness of our integrated solution.

Exploring Alternative Approaches

While integration by parts is the most straightforward approach, let's briefly explore alternative methods, although they might be less efficient in this specific case.

Substitution Method

The substitution method isn't directly applicable here because there's no obvious substitution that simplifies the integral significantly. Attempting substitution would lead to more complex expressions rather than simplification.

Tabular Integration

Tabular integration is a variation of integration by parts, especially useful when repeated applications are required. While it can be used for this problem, it doesn't offer a significant advantage over the standard integration by parts method in this instance.

Practical Applications and Further Exploration

The integral of xlnx, while seemingly a theoretical exercise, has applications in various fields. It appears in probability density functions, particularly in problems involving logarithmic distributions. Understanding its solution provides a building block for more complex integrations encountered in advanced calculus and related disciplines.

Furthermore, exploring the integral of similar functions, such as x²lnx, x³lnx, or even more generalized forms involving higher powers of x, can deepen your understanding of integration techniques and the interplay between algebraic and logarithmic functions. These exercises reinforce the mastery of integration by parts and expand problem-solving capabilities. Each additional power of x introduces an extra layer to the integration by parts process, necessitating repeated applications of the technique.

Remember that consistent practice and exploring various examples are key to mastering integration techniques. The more you practice, the more intuitively you'll choose the optimal approach for each integral you encounter.

Conclusion

The integral of xlnx provides a valuable and practical illustration of integration by parts, a fundamental technique in calculus. Mastering this integral, alongside understanding the underlying principles and exploring alternative approaches, solidifies your foundation in integration and prepares you for more complex integration problems. Remember to always verify your solution through differentiation to ensure accuracy and to enhance your understanding of the relationship between integration and differentiation. The beauty of calculus lies in its interconnectedness, and understanding this relationship is key to mastering the subject.