Understanding the Integral of 1/xlnx - Step by Step

Let's analyze a challenging integral that requires careful consideration.We're faced with the integral of one over x times natural log of x.To understand this integral better, let's break down its components.In the numerator, we simply have 1, a constant. The denominator contains two terms: x and the natural logarithm of x.This integral has several key characteristics that make it special.Before we attempt to solve this, we need to consider the domain restrictions.X must be greater than zero for the natural logarithm to be defined, and x cannot equal 1 because that would make the denominator zero.The presence of both x and natural log of x in the denominator suggests we should look for a substitution involving the natural logarithm.Now that we understand the structure of our integral, we can move on to solving it.To solve this integral, we'll use the substitution u equals ln x.This means that d u equals one over x d x.We can rearrange this to x d u equals d x, which will help us substitute.Since u equals ln x, the inverse relationship tells us that x equals e to the u.Let's carefully substitute each part of our integral.One over x ln x becomes one over e to the u times u, and dx becomes e to the u du.When we put this all together, our integral transforms into this new form.Which simplifies to the integral of e to the negative u over u with respect to u.Notice how the e to the u from our substitution of dx combines with the e to the negative u in the denominator, simplifying our expression.This substitution has transformed our original complex integral into this more manageable form.From our previous substitution, we now have this integral.We can recognize that this form will give us the natural log of the absolute value of ln x plus C.To verify this is correct, let's differentiate our proposed answer using the chain rule.Starting with y equals ln absolute value of ln x plus CUsing the chain rule, we get one over ln x times one over xWhich simplifies to one over x ln x, exactly what we started with!However, we need to consider some important domain restrictions.For this integral to be valid, x must be greater than zero for the natural log to be defined.And x cannot equal one, because ln of one equals zero, which would make us divide by zero.Therefore, our final answer is the integral of one over x ln x equals ln absolute value of ln x plus C.This completes our solution to this challenging integral.