Question

What is the solution of \( \int x^2 \ln x \, dx \)? Given \( C \) is an arbitrary constant. 
 

• A. \( \frac{x^3}{3} \ln x - \frac{x^3}{9} + C \)
• B. \( \frac{x^3}{3} \ln x + \frac{x^3}{9} + C \)
• C. \( \frac{x^3}{9} \ln x - \frac{x^3}{3} + C \)
• D. \( \frac{x^3}{9} \ln x - \frac{x^3}{3} + C \)

Hint:

Use integration by parts for integrals involving a product of a polynomial and a logarithmic function.

Answer 1

The Correct Option is A

Step 1: Integration by parts.
To solve \( \int x^2 \ln x \, dx \), we apply integration by parts. Let: \[ u = \ln x \text{and} dv = x^2 \, dx \] Then, \( du = \frac{1}{x} \, dx \) and \( v = \frac{x^3}{3} \).

Step 2: Applying the formula.
The integration by parts formula is: \[ \int u \, dv = uv - \int v \, du \] Substitute the values: \[ \int x^2 \ln x \, dx = \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx \] Simplifying the second integral: \[ = \frac{x^3}{3} \ln x - \frac{1}{3} \int x^2 \, dx = \frac{x^3}{3} \ln x - \frac{x^3}{9} + C \]

Step 3: Conclusion.
The correct answer is (A) \( \frac{x^3}{3} \ln x - \frac{x^3}{9} + C \).