Antiderivative of x^4/(5 - x^5): A Step-by-Step Guide

This article provides a detailed explanation of how to solve the antiderivative of x^4 divided by 5 - x^5. The process involves the method of integration by substitution, which is a fundamental technique in calculus. Let's break down the solution step by step and explore its implications and applications.

Introduction to the Problem

The problem at hand is to find the antiderivative, or integral, of the following function:

[int frac{x^4}{5 - x^5} , dx]

This integral is challenging because the direct approach doesn't fit neatly with standard techniques like u-substitution, which often require matching the derivative of the denominator to a term in the numerator. However, with careful analysis and substitution, we can simplify this integral significantly.

Identifying the Correct Substitution

To solve this integral, we look at the denominator 5 - x^5. To simplify the expression, we notice that the derivative of x^5 is 5x^4, and indeed, x^4 is present in the numerator. This observation indicates that we can use the substitution:

[u 5 - x^5]

This substitution simplifies the integral significantly because we need to incorporate the differential du. By differentiating both sides of [u 5 - x^5], we obtain:

[du -5x^4 , dx]

Rearranging this, we get:

[x^4 , dx -frac{1}{5} , du]

Substitution into the Integral

Now, we can substitute u and du into the original integral:

[int frac{x^4}{5 - x^5} , dx int left( -frac{1}{5} right) frac{1}{u} , du -frac{1}{5} int frac{1}{u} , du]

The integral of [frac{1}{u}] is [ln|u|], so:

[-frac{1}{5} int frac{1}{u} , du -frac{1}{5} ln|u| C]

Substituting back [u 5 - x^5], we obtain the final result:

[-frac{1}{5} ln|5 - x^5| C]

Additional Insights and Applications

This integral has many applications in various fields, including physics and engineering, where it may be used to determine areas under curves or in solving kinematic problems involving changing forces or velocities. Understanding how to solve such integrals is crucial for students and professionals alike.

Implications of the Solution

The solution to the integral highlights the importance of recognizing patterns and choosing the right substitution in complex integrations. The method shown here, while straightforward, requires a keen eye for detail and a strong foundation in calculus.

Conclusion

In summary, the antiderivative of x^4 divided by 5 - x^5 is given by:

[-frac{1}{5} ln|1 - x^5| C]

This detailed explanation should provide a clear understanding of the integration process and its application. Whether you are a student, a teacher, or a professional, mastering such techniques is a valuable skill in the realm of calculus.