Question

Find the value of: \[ \int \frac{1}{x} \, dx \]

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

Hint:

Remember that the power rule of integration fails only when the exponent is \( -1 \).
In that special case, the integral always transitions to a logarithmic function.
This is one of the most frequently tested concepts in entrance exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem asks for the indefinite integral of the rational function \( f(x) = \frac{1}{x} \) with respect to \( x \).
We need to identify the antiderivative of this function from standard calculus integration formulas.

Step 2: Key Formula or Approach:
The power rule of integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) \]
When \( n = -1 \), the expression becomes \( x^{-1} = \frac{1}{x} \).
Using the standard power rule here would lead to division by zero, which is undefined.
Therefore, we must use the specific logarithmic integration rule:
\[ \int \frac{1}{x} \, dx = \ln|x| + C \]
In many contexts, particularly in standard textbooks and exams, \( \ln x \) is represented simply as \( \log x \).

Step 3: Detailed Explanation:
We are seeking a function \( F(x) \) such that its derivative \( F'(x) = \frac{1}{x} \).
From differential calculus, we know that the derivative of the natural logarithmic function is:
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]
By the fundamental theorem of calculus, the inverse process of differentiation is integration.
Thus, the antiderivative of \( \frac{1}{x} \) must be \( \log x + C \), where \( C \) is the constant of integration.
Let us evaluate the other options:
- Option (B) is incorrect because \( \frac{d}{dx}(\frac{1}{x^2}) = -2x^{-3} \), which is not \( \frac{1}{x} \).
- Option (C) is incorrect because \( \frac{d}{dx}(x\log x) = \log x + 1 \) by using the product rule.
- Option (D) is incorrect because the derivative of \( e^x \) is \( e^x \), not \( \frac{1}{x} \).

Step 4: Final Answer:
Therefore, the correct evaluation of the integral is \( \log x + C \), which corresponds to Option (A).