Question

If \(\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C\), where C is constant of integration, then f(x) is

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

Hint:

Whenever you see an integral involving \(e^x\) multiplied by a function, always try to check if the function can be expressed as the sum of another function and its derivative, i.e., \(g(x) + g'(x)\). This shortcut can save a lot of time compared to using integration by parts.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves integration by recognizing a specific form. The integral form \(\int e^x (g(x) + g'(x)) dx = e^x g(x) + C\) is a standard and very useful result derived from the product rule of differentiation. The goal is to manipulate the given integrand into this form.
Step 2: Key Formula or Approach:
The key formula to use is: \[ \int e^x (g(x) + g'(x)) dx = e^x g(x) + C \] We need to rewrite the given integral to match this pattern.
Step 3: Detailed Explanation:
The given integral is: \[ I = \int \frac{(1 + x \log x)}{xe^{-x}} dx \] First, let's simplify the integrand. The term \(e^{-x}\) in the denominator is equivalent to \(e^x\) in the numerator. \[ I = \int e^x \frac{(1 + x \log x)}{x} dx \] Now, let's split the fraction inside the integral: \[ I = \int e^x \left(\frac{1}{x} + \frac{x \log x}{x}\right) dx \] \[ I = \int e^x \left(\frac{1}{x} + \log x\right) dx \] Now we check if this expression fits the form \(\int e^x (g(x) + g'(x)) dx\).
Let's try setting \(g(x) = \log x\).
If \(g(x) = \log x\), then its derivative is \(g'(x) = \frac{1}{x}\).
Substituting these into our integral, we get: \[ I = \int e^x (g'(x) + g(x)) dx \] This perfectly matches the standard form.
Therefore, the result of the integration is: \[ I = e^x g(x) + C = e^x \log x + C \] The problem states that the integral is equal to \(e^x f(x) + C\).
By comparing our result \(e^x \log x + C\) with the given form, we can see that: \[ f(x) = \log x \] Step 4: Final Answer:
The function f(x) is \(\log x\).