What Is The Integral Of Xlnx

Solving the Integral of xlnx: A Step-by-Step Guide

This article will guide you through the process of solving the indefinite integral of xlnx, a common problem encountered in calculus. We'll break down the solution using integration by parts, explaining each step clearly. Understanding this integral is crucial for various applications in mathematics, physics, and engineering. This guide will also cover related concepts and potential pitfalls.

Understanding Integration by Parts

Before diving into the solution, let's refresh our understanding of integration by parts. This technique is essential for integrating functions that are products of two simpler functions. The formula is:

∫u dv = uv - ∫v du

Choosing the right 'u' and 'dv' is key to simplifying the integral. A useful mnemonic is "LIATE": Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential. This order generally suggests the best choice for 'u'.

Solving ∫xlnx dx

1. Choosing u and dv:

   In our case, we have the function xlnx. Following the LIATE rule, we choose:

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

2. Finding du and v:

   Now we differentiate 'u' to find 'du':

   • du = (1/x) dx

   And we integrate 'dv' to find 'v':

   • v = ∫x dx = (x²/2)

3. Applying the Integration by Parts Formula:

   Substitute our values of u, v, du, and dv into the integration by parts formula:

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

4. Simplifying and Integrating:

   Simplify the remaining integral:

   ∫xlnx dx = (x²ln x)/2 - ∫(x/2) dx

   Now, integrate the simplified expression:

   ∫xlnx dx = (x²ln x)/2 - (x²/4) + C

The Final Result

Therefore, the indefinite integral of xlnx is:

(x²ln x)/2 - (x²/4) + C

Where 'C' is the constant of integration. Remember to always include the constant of integration when solving indefinite integrals.

Common Mistakes to Avoid

• Incorrect choice of 'u' and 'dv': Choosing the wrong functions for 'u' and 'dv' can significantly complicate the integration process. The LIATE rule can be a helpful guideline, but practice is key to mastering this skill.
• Forgetting the constant of integration: Always remember to add the constant of integration ('C') at the end of your solution. This accounts for all possible antiderivatives of the function.
• Errors in differentiation and integration: Pay close attention to the rules of differentiation and integration to avoid errors in calculating 'du' and 'v'.

Practical Applications

Understanding how to solve integrals like ∫xlnx dx is crucial for solving various problems in different fields, including:

• Probability and Statistics: Used in calculating expected values and other statistical measures.
• Physics: Appears in calculations related to work, energy, and other physical quantities.
• Engineering: Used in solving differential equations and modeling various systems.

By mastering integration by parts, you'll unlock a powerful tool for solving a wide array of calculus problems. Remember to practice consistently and carefully review each step to avoid errors. This detailed explanation should provide a solid foundation for understanding and solving similar integrals in the future.