Integration By Parts Log X: Worked Example and Solution

Problem 1: Evaluate $\int\ln x\,dx$.

Answer: $x\ln x-x+C$

Why this method: Integration by parts matches the mathematical structure before any algebraic cleanup.

1. Write the integrand as $(\ln x)(1)$; take $u=\ln x$ and $dv=dx$, so $du=dx/x$ and $v=x$.
2. Then $I=x\ln x-\int1\,dx=x\ln x-x+C$, for $x>0$.
3. Therefore the result is $x\ln x-x+C$.