Evaluating The Integral Of X³ Ln(x) Dx A Step-by-Step Guide

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This article provides a comprehensive guide on how to evaluate the integral ∫x³ ln(x) dx. This integral is a classic example that requires the use of integration by parts, a fundamental technique in calculus. We will break down the process step-by-step, explaining the underlying principles and demonstrating the application of the formula. The discussion will cover the selection of u and dv, the calculation of du and v, and the final steps to arrive at the solution. Mastering this technique will significantly enhance your ability to solve a wide range of integrals involving products of functions.

Understanding Integration by Parts

Before diving into the specific integral, it's crucial to understand the concept of integration by parts. This technique is derived from the product rule of differentiation and is particularly useful when integrating a product of two functions. The formula for integration by parts is:

∫u dv = uv - ∫v du

Where u and v are functions of x, and du and dv are their respective differentials. The key to successfully applying integration by parts lies in choosing the appropriate functions for u and dv. The goal is to select u such that its derivative, du, simplifies the integral, and dv such that it can be easily integrated to find v. A common mnemonic to aid in this selection is LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. This acronym suggests the order of preference for choosing u: if the integral contains a logarithmic function and an algebraic function, the logarithmic function should generally be chosen as u.

The essence of integration by parts is to transform a complex integral into a simpler one. By carefully selecting u and dv, the integral ∫v du is often easier to evaluate than the original integral ∫u dv. This process may even need to be applied multiple times to reach a solvable integral, especially when dealing with more complicated functions. The ability to recognize when and how to apply integration by parts is a fundamental skill in integral calculus.

Step-by-Step Solution for ∫x³ ln(x) dx

Let's now apply the integration by parts technique to evaluate the integral ∫x³ ln(x) dx. Following the LIATE rule, we identify ln(x) as the logarithmic function and x³ as the algebraic function. Therefore, we choose:

  • u = ln(x)
  • dv = x³ dx

Next, we need to find du and v. The derivative of u = ln(x) is:

  • du = (1/x) dx

And the integral of dv = x³ dx is:

  • v = ∫x³ dx = (x⁴)/4

Now, we can substitute these values into the integration by parts formula:

∫x³ ln(x) dx = ln(x) * (x⁴)/4 - ∫(x⁴)/4 * (1/x) dx

Simplifying the expression, we get:

∫x³ ln(x) dx = (x⁴ ln(x))/4 - (1/4) ∫x³ dx

The remaining integral ∫x³ dx is a simple power rule integral, which we can easily evaluate:

∫x³ dx = (x⁴)/4 + C

Where C is the constant of integration. Substituting this back into the equation, we have:

∫x³ ln(x) dx = (x⁴ ln(x))/4 - (1/4) * (x⁴)/4 + C

Finally, simplifying the expression gives us the solution:

∫x³ ln(x) dx = (x⁴ ln(x))/4 - (x⁴)/16 + C

This step-by-step approach demonstrates how integration by parts can be used to solve integrals involving products of functions. By carefully selecting u and dv and applying the formula, we can break down complex integrals into simpler ones.

Choosing u and dv: A Crucial Step

As highlighted earlier, the selection of u and dv is a critical step in applying integration by parts. A poor choice can lead to a more complicated integral than the original, while a judicious choice simplifies the problem significantly. The LIATE mnemonic provides a useful guideline, but it's not a rigid rule. Sometimes, you might need to deviate from this order to achieve the simplest solution. For example, if your integral involves an inverse trigonometric function multiplied by an algebraic function, LIATE suggests choosing the inverse trigonometric function as u. This is because the derivative of an inverse trigonometric function is often simpler than the function itself, whereas the integral of an algebraic function is also relatively straightforward.

Consider the integral ∫x * e^x dx. Following LIATE, the algebraic function x would be chosen as u, and the exponential function e^x would be dv. This choice leads to du = dx and v = e^x. Applying integration by parts yields:

∫x * e^x dx = x * e^x - ∫e^x dx = x * e^x - e^x + C

This example illustrates the effectiveness of LIATE in many cases. However, it's important to remember that the ultimate goal is to simplify the integral. If following LIATE leads to a more complex expression, it's worth considering alternative choices for u and dv. Practice and experience are the best ways to develop intuition for making the right choice.

Common Mistakes and How to Avoid Them

When applying integration by parts, there are several common mistakes that students often make. One frequent error is incorrectly calculating the derivative du or the integral v. Double-checking these calculations is crucial to avoid propagating errors throughout the solution. Another common mistake is forgetting the constant of integration, C, in the final answer. While this may seem like a minor oversight, it's important to remember that the indefinite integral represents a family of functions, all differing by a constant.

Another potential pitfall is choosing u and dv in a way that complicates the integral. As discussed earlier, the LIATE mnemonic provides a helpful guideline, but it's not foolproof. If the integral ∫v du is more complex than the original integral ∫u dv, you may need to reconsider your choice of u and dv. Sometimes, you may even need to apply integration by parts multiple times to reach a solvable integral. For example, consider the integral ∫x² * sin(x) dx. Applying integration by parts once will reduce the power of x, but you'll still have an integral involving a product of functions. Applying integration by parts a second time will eliminate the x term altogether, leading to a straightforward integral.

Finally, it's important to be meticulous with notation and algebraic manipulations. A misplaced sign or an incorrect simplification can easily lead to an incorrect answer. Taking the time to write out each step clearly and carefully can help minimize these errors. Regularly practicing integration by parts with a variety of examples is the best way to develop proficiency and avoid common mistakes.

Advanced Techniques and Applications

While the basic integration by parts formula is powerful, there are situations where more advanced techniques or variations are required. One such technique is tabular integration, also known as the