Integration By Parts Log X: Worked Example and Solution
A complete integration by parts example with method choice, derivation, verification, and a common wrong approach.
1 problem8 min estimated timeIntermediate to advanced progression
What is included
See a concise answer and full derivation for evaluate \(\int\ln x\,dx\).
Skills assessed
- Integration by parts
Prerequisites
- antiderivatives
- product rule
Long description
Read the diagram from top to bottom. Each numbered box names one decision or mathematical operation. Arrows show the required order; the text labels remain the complete interpretation in print, dark mode, and nonvisual reading.
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Printable preview
- Evaluate .
Complete worked solutions
Every problem has a source-matched answer and independently reviewed derivation.
01
Problem 1: Evaluate .
Answer:
Why this method: Integration by parts matches the mathematical structure before any algebraic cleanup.
- Choose u as the factor that simplifies when differentiated and integrate dv.
- Apply ; repeat when the remaining integral still has a polynomial factor.
- Simplification gives .
Common errors
- Choosing dv that is harder to integrate or losing the minus sign in the formula.