What is the integration of 1/(a-x) dx?

Calculus, a branch of mathematics that deals with the study of change, is widely used in various fields from physics to economics. One fundamental concept in calculus is integration, which helps in finding the area under a curve or solving complex problems. In this article, we will delve into a specific integration problem: the integration of 1/(a-x) dx. We will provide a step-by-step guide on how to solve this problem, and discuss its relevance in the realm of calculus. Let's begin our journey!

Understanding the Problem

We are given the integral:

∫ 1/(a-x) dx

Understanding the integral is the first step in solving it. The problem involves finding the antiderivative of the function 1/(a-x) with respect to x.

Solving the Integration

To solve ∫ 1/(a-x) dx, we can use a technique called substitution. By setting a variable equal to a simpler form of the given expression, we can transform the integral into a more manageable form. Let's explore this technique step-by-step.

Step-by-Step Guide

Step 1: Assume a-x z. This substitution simplifies the expression inside the integral.

Step 2: Differentiate both sides with respect to x: dx -dz. This step is crucial as it transforms the differential dx in the original integral into a new differential -dz.

Step 3: Substitute these expressions into the original integral. The integral becomes:

∫ 1/(a-x) dx ∫ -1/z dz

This new integral is simpler to solve as it is now in terms of z.

Step 4: Integrate with respect to z. The integral of -1/z dz is:

-ln|z| C, where C is the constant of integration.

Step 5: Substitute back the original variable x in place of z. Since z a-x, we get:

-ln|a-x| C

This is the final result of the integration, as shown here:

∫ 1/(a-x) dx -ln|a-x| C

Conclusion

In conclusion, solving integrals like 1/(a-x) dx can be approached through substitution, making complex integrals more manageable. Understanding the steps involved in substitution and integrating simple expressions is crucial for tackling more complex problems in calculus.

Whether you're a student of mathematics, an engineer, or someone interested in the applications of calculus, mastering integrals can greatly enhance your problem-solving skills. If you're looking to explore more calculus concepts or need help with similar problems, consider consulting online resources or seeking guidance from a tutor.

Related Keywords

Integration Calculus Definite Integration

References

For further reading and a deeper understanding of integration and calculus, refer to renowned calculus texts or explore online resources such as Khan Academy's Calculus 1 and