Integration of 1/(5x-4) with Step-by-Step Guide

In this article, we will explore the process of integrating the function 1/(5x-4). Integration is a fundamental concept in calculus, used extensively in various mathematical and scientific applications. We will provide a detailed step-by-step guide on how to integrate this function and include practical examples.

Introduction to Integration

Integration is the process of finding the integral of a function, which can be broadly classified into two types: definite and indefinite integrals. The definite integral of a function represents the area under the curve over a specific interval, whereas the indefinite integral provides the antiderivative of the function, which includes a constant of integration.

Integration Techniques

Several techniques can be used to integrate functions, including substitution, partial fractions, and trigonometric identities. In this article, we will focus on a common technique for finding the antiderivative of rational functions using substitution.

Step-by-Step Integration of 1/(5x-4)

Let's consider the function 1/(5x-4). The goal is to find its antiderivative. We will follow these steps:

Substitution: To simplify the integration, we use the substitution method. Let u 5x - 4. This substitution will help us in simplifying the integrand.

Calculate the differential: Next, we need to find the differential of u. We have:

du d(5x - 4) 5dx

Thus, dx du/5.

Rewrite the integral: Substitute u and dx into the original integral:

∫ 1/(5x-4) dx ∫ 1/u (du/5) (1/5) ∫ 1/u du

Integrate: The integral of 1/u is ln|u| C. Therefore, we have:

(1/5) ∫ 1/u du (1/5) ln|u| C

Substitute back: Finally, substitute u 5x - 4 back into the expression:

(1/5) ln|5x - 4| C

The final answer is:

(1/5) ln|5x - 4| C

Common Pitfalls and Misinterpretations

It is important to be clear and precise when writing mathematical expressions. In the original question, the expression 1/(5x-4) dx was ambiguous, leading to different interpretations of the integral. To avoid such ambiguities, it is crucial to use parentheses and clearly specify the integrand and the variable of integration.

Example Integrals

Let's consider a couple of additional examples to demonstrate the process of integration:

∫ x/(5x - 4) dx

(1/5) ∫ x dx - ∫ 4 dx (1/5)(x^2/2) - 4x C

x^2/10 - 4x C

∫ 1/(5x - 4) dx

(1/5) ln|5x - 4| C

∫ 1/((5x - 4) dx

(1/5) ln|5x - 4| C (assuming the intended operation)

This article provides a comprehensive guide to integrating the function 1/(5x-4). By understanding and applying the integration techniques discussed, you will be well-equipped to handle similar problems in your own studies and applications.