Integral of ln(x) * x^(ln(x)-1): A Step-by-Step Guide

In this article, we will explore the integral of the function ln(x) * x^(ln(x)-1). This integral is commonly encountered in advanced calculus and has significant applications in various fields of mathematics and science. We will walk through the solution methodically, step-by-step, using multiple techniques including integration by substitution.

Introduction to the Problem

The integral in question is: $$ int ln(x) cdot x^{ln(x)-1} , dx $$ This integral requires a series of algebraic manipulations and application of integration techniques to solve.

Solution Method

First, let's rewrite the integral in a more convenient form: $$ int x^{ln(x)} cdot ln(x) cdot frac{1}{x} , dx $$ By recognizing that $ x^{ln(x)} e^{(ln(x))^2} $, we proceed to the next step.

Method 1: Direct Integration by Substitution

Let's use the substitution ( u ln(x) ). Then, ( du frac{1}{x} dx ).

The integral becomes: $$ int e^{u^2} cdot u cdot x , du $$ Using another substitution ( alpha u^2 ), we get ( dalpha 2u du ) or ( du frac{dalpha}{2u} ).

Substituting back, we get: $$ int frac{1}{2} e^{alpha} , dalpha frac{1}{2} e^{alpha} C $$ Reversing the substitution ( alpha u^2 ), we get: $$ frac{1}{2} e^{u^2} C frac{1}{2} e^{(ln(x))^2} C frac{1}{2} x^{ln(x)} C $$ Thus, the integral is: $$ int ln(x) cdot x^{ln(x)-1} , dx frac{1}{2} x^{ln(x)} C $$

Method 2: Simplified Integration by Substitution

Alternatively, let's directly substitute ( u x^{ln(x)} ). Then, ( du 2x^{ln(x)-1} ln(x) , dx ) or ( frac{du}{2 ln(x)} x^{ln(x)-1} ln(x) , dx ).

The integral simplifies to: $$ int frac{1}{2} , du frac{1}{2} u C frac{1}{2} x^{ln(x)} C $$

Verification

To verify our solution, we differentiate the result and check if we get the original integrand: $$ frac{d}{dx} left( frac{1}{2} x^{ln(x)} right) frac{1}{2} left( x^{ln(x)} ln(x) x^{ln(x)-1} right) frac{1}{2} x^{ln(x)} left( ln(x) frac{1}{x} right) frac{1}{2} x^{ln(x)} cdot frac{2 ln(x) x}{2x} ln(x) cdot x^{ln(x)-1} $$

Conclusion

In this article, we explored the integral of ln(x) * x^(ln(x)-1) and solved it using two different methods. Both methods lead to the same result:

$$ int ln(x) cdot x^{ln(x)-1} , dx frac{1}{2} x^{ln(x)} C $$

This integral is an important example in calculus, demonstrating the power of integration by substitution and logarithmic differentiation techniques. Understanding these methods is crucial for solving more complex integrals.