Q.21
What is the solution of \( \int \frac{x}{x^2 \ln x} \, dx \)? Given \( C \) is an arbitrary constant.
Evaluating the integration of x squared ln x is a classic calculus question that frequently appears in competitive exams and university tests. This problem not only checks familiarity with integration by parts but also the ability to verify each multiple-choice option by differentiation. Understanding the complete solution and analyzing every option strengthens conceptual clarity and reduces errors in timed exams.
Step-by-Step Solution
Use integration by parts: \(\int u\,dv = uv – \int v\,du\)
Choose:
- \(u = \ln x \Rightarrow du = \frac{1}{x}dx\)
- \(dv = x^2 dx \Rightarrow v = \frac{x^3}{3}\)
$$= \frac{x^3}{3}\ln x – \frac{1}{3} \cdot \frac{x^3}{3} + C = \frac{x^3}{3}\ln x – \frac{x^3}{9} + C$$
Option Analysis
Let \(F(x)\) be each option and differentiate to verify \(F'(x) = x^2 \ln x\).
Option (A): \(\frac{x^3}{3}\ln x – \frac{x^3}{9} + C\)
Differentiate: \(d/dx\left(\frac{x^3}{3}\ln x\right) = x^2\ln x + x^2\)
\(d/dx\left(-\frac{x^3}{9}\right) = -\frac{x^2}{3}\)
Total: \(x^2\ln x + x^2 – \frac{x^2}{3} – \frac{2x^2}{3} = x^2\ln x\) ✅ Correct
Option (B): \(\frac{x^3}{3}\ln x + \frac{x^3}{9} + C\)
Differentiate: \(d/dx\left(\frac{x^3}{3}\ln x\right) = x^2\ln x + x^2\)
\(d/dx\left(\frac{x^3}{9}\right) = \frac{x^2}{3}\)
Total: \(x^2\ln x + x^2 + \frac{x^2}{3} = x^2\ln x + \frac{4x^2}{3}\) ❌ Incorrect
Option (C): \(-\frac{x^3}{9}\ln x + \frac{x^3}{9} + C\)
Differentiate: \(d/dx\left(-\frac{x^3}{9}\ln x\right) = -\left(\frac{x^2}{3}\ln x + \frac{x^2}{3}\right)\)
Total: \(-\frac{x^2}{3}\ln x – \frac{x^2}{3} + \frac{x^2}{3} = -\frac{x^2}{3}\ln x\) ❌ Incorrect
Option (D): \(\frac{x^3}{9}\ln x – \frac{x^3}{3} + C\)
Differentiate: \(d/dx\left(\frac{x^3}{9}\ln x\right) = \frac{x^2}{3}\ln x + \frac{x^2}{3}\)
Total: \(\frac{x^2}{3}\ln x + \frac{x^2}{3} – x^2 = \frac{x^2}{3}\ln x – \frac{2x^2}{3}\) ❌ Incorrect
Final Answer
Among the given options, Option (A) is the correct antiderivative that differentiates back to \(x^2 \ln x\).
Exam Tips
- Always verify MCQ options by differentiation, not just integration
- Follow ILATE rule for choosing \(u\) in integration by parts
- Watch for sign errors in the constant term
- Practice similar problems: \(\int x^n \ln x\,dx\)
