Question

Let \[ f(x)=\int\left(\frac{1}{\log_e x}-\frac{2}{(\log_e x)^3}\right)\,dx. \] If \(f(e)=2e\), then \(f(e^2)\) is equal to:

• A. \(\dfrac{e^2}{4}\)
• B. \(\dfrac{e^2}{2}\)
• C. \(\dfrac{3e^2}{4}\)
• D. \(\dfrac{4e^2}{3}\)

Hint:

When an integrand looks complicated, try differentiating expressions of the form \(x(\log x)^n\) or \(x/(\log x)^n\) to match terms efficiently.

Answer 1

The Correct Option is C

Concept:
Try to identify the integrand as the derivative of a suitable expression.
Logarithmic differentiation is helpful when integrands involve powers of \(\log x\).
Use the given value of the function to determine the constant of integration.
Step 1: Observe the integrand Consider the function \[ F(x)=x\left(\frac{1}{\log x}+\frac{1}{(\log x)^2}\right) \] Differentiate: \[ F'(x)=\left(\frac{1}{\log x}+\frac{1}{(\log x)^2}\right) +x\left(-\frac{1}{x(\log x)^2}-\frac{2}{x(\log x)^3}\right) \] \[ F'(x)=\frac{1}{\log x}-\frac{2}{(\log x)^3} \] Hence, \[ f(x)=x\left(\frac{1}{\log x}+\frac{1}{(\log x)^2}\right)+C \]
Step 2: Use the given condition \(f(e)=2e\) \[ f(e)=e\left(1+1\right)+C=2e+C \] Given \(f(e)=2e\), \[ C=0 \]
Step 3: Find \(f(e^2)\) \[ f(e^2)=e^2\left(\frac{1}{2}+\frac{1}{4}\right) =e^2\cdot\frac{3}{4} =\frac{3e^2}{4} \]