Introduction

In calculus, finding the antiderivative of a function is a fundamental operation. This article will delve into the antiderivatives of two important functions, ( ln(x) ) and ( e^x ), using methods such as integration by parts and direct integration. We will also explore the relationship between ( ln(x) ) and exponential functions.

Integration by Parts for ( ln(x) )

First, consider the antiderivative of ( ln(x) ). Using the integration by parts formula, which states that ( int u , dv uv - int v , du ), we can find the antiderivative.

The Process:

Let ( u ln(x) ) and ( dv dx ). Then, ( du frac{1}{x} , dx ) and ( v x ). Substituting these values into the integration by parts formula:

$$int ln(x) , dx xln(x) - int x cdot frac{1}{x} , dx$$

$$int ln(x) , dx xln(x) - int 1 , dx$$

$$int ln(x) , dx xln(x) - x C$$

Therefore, the antiderivative of ( ln(x) ) is ( xln(x) - x C ).

Direct Integration for ( e^x )

The antiderivative of ( e^x ) is straightforward. By directly integrating, we know that:

$$int e^x , dx e^x C$$

This is because the derivative of ( e^x ) is ( e^x ), confirming that ( e^x ) is its own antiderivative.

Exponential Functions and Logarithms

The expression ( ln(e^x) ) simplifies to ( x ) because the natural logarithm function ( ln(x) ) is the inverse of the exponential function ( e^x ). Therefore:

$$ln(e^x) x$$

This relationship provides a useful tool for simplifying expressions involving exponentials and logarithms.

Application in Real-world Problems

Understanding the antiderivatives of ( ln(x) ) and ( e^x ) is crucial in various fields such as physics, engineering, and economics. For instance, in physics, ( e^x ) can be used to model exponential growth in population or radioactive decay. In economics, ( ln(x) ) can be used in growth calculations.

Conclusion

In summary, the antiderivatives of ( ln(x) ) and ( e^x ) are essential components of calculus. Using integration by parts for ( ln(x) ) and direct integration for ( e^x ) provides a powerful set of tools for solving a wide range of mathematical and real-world problems.