Question

If \(\int x \log x \, dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\), then \( f(x) \) is

• A. \((\log x)^{-1}\)
• B. \(2\log x\)
• C. \(\log x\)
• D. \(3\log x\)

Hint:

Remember the LIATE rule for choosing 'u' in integration by parts. This priority order (Logarithmic>Inverse>Algebraic>Trigonometric>Exponential) simplifies the process and helps avoid complicated integrals.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires us to find the function f(x) by evaluating the integral \(\int x \log x \,dx\) and comparing it with the given expression. The integral can be solved using the method of integration by parts.
Step 2: Key Formula or Approach:
The formula for integration by parts is: \[ \int u \,dv = uv - \int v \,du \] To choose 'u' and 'dv', we use the LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential). The function that comes first in this order should be chosen as 'u'.
Step 3: Detailed Explanation:
We need to evaluate \(I = \int x \log x \,dx\).
According to the LIATE rule, we choose the logarithmic function as 'u' and the algebraic function as 'dv'.
Let \(u = \log x\) and \(dv = x \,dx\).
Then, we differentiate 'u' and integrate 'dv': \[ du = \frac{1}{x} \,dx \] \[ v = \int x \,dx = \frac{x^2}{2} \] Now, applying the integration by parts formula: \[ I = (\log x)\left(\frac{x^2}{2}\right) - \int \left(\frac{x^2}{2}\right)\left(\frac{1}{x}\right) \,dx \] \[ I = \frac{x^2}{2} \log x - \int \frac{x}{2} \,dx \] \[ I = \frac{x^2}{2} \log x - \frac{1}{2} \int x \,dx \] \[ I = \frac{x^2}{2} \log x - \frac{1}{2} \left(\frac{x^2}{2}\right) + c \] \[ I = \frac{x^2}{2} \log x - \frac{x^2}{4} + c \] The problem states that \(\int x \log x \,dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\).
Comparing our result with the given equation: \[ \frac{x^2}{2} \log x - \frac{x^2}{4} + c = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c \] By comparing the terms, we can see that: \[ \frac{x^2}{2} f(x) = \frac{x^2}{2} \log x \] \[ f(x) = \log x \] Step 4: Final Answer:
The function f(x) is \(\log x\). So, option (C) is correct.