Integration of eax: A Comprehensive Guide

This article provides a detailed explanation of the integration of the exponential function eax with respect to x. We will explore multiple methods to understand and solve this integral, ensuring you have a solid grasp of the concept and its applications.

Introduction to the Problem

The integral of the function fx eax with respect to x is a common problem in calculus.

Using Substitution

One straightforward method to solve the integral is using the substitution method. Let's assume:

ax t

The differential of this expression is:

dx 1/adt

Substituting these into the integral:

∫ et 1/adt 1/a∫ etdt et C

Since t ax, the final answer is:

1/aeax C

Direct Integration

Another effective method is to directly apply the integration formula for eax. The formula is:

∫ eax dx 1/aeax C

Using Integration by Parts

Let's consider using integration by parts. We choose:

U x du dx dv eaxdx v 1/aeax

Applying the integration by parts formula:

Integral UdV UV - Integral VDU

Substituting the values:

∫ eaxdx x1/aeax- ∫ 1/aeaxdx

Solving further:

x1/aeax-1/a2eax C

Reducing and organizing:

x1/aeax-1/a2eax C

This is the final integrated form.

Conclusion

Understanding the integration of eax is crucial for any calculus student and mathematician. This article has covered the method of substitution, direct integration, and integration by parts to help you solve the integral effectively.