Question

$$ \int \frac{1}{(x-3)(x-7)} \, dx = ? $$

• A. $\frac{1}{4} \log \left( \frac{x+7}{x+3} \right) + c$
• B. $\frac{1}{4} \log \left( \frac{x-7}{x+3} \right) + c$
• C. $\frac{1}{4} \log \left( \frac{x-7}{x-3} \right) + c$
• D. None

Hint:

Always subtract the smaller constant from the larger constant ($7 - 3$) to determine the coefficient in front of the logarithm.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is an integral of a rational function that can be solved by decomposing the fraction into simpler parts using partial fractions.

Step 2: Key Formula or Approach:
Use the standard formula: $\int \frac{dx}{(x-a)(x-b)} = \frac{1}{a-b} \log \left| \frac{x-a}{x-b} \right| + c$.

Step 3: Detailed Explanation:
1. Express the fraction as a difference: \[ \frac{1}{(x-7)(x-3)} = \frac{1}{7-3} \left( \frac{1}{x-7} - \frac{1}{x-3} \right) = \frac{1}{4} \left( \frac{1}{x-7} - \frac{1}{x-3} \right) \] 2. Integrate each term: \[ \frac{1}{4} \int \frac{dx}{x-7} - \frac{1}{4} \int \frac{dx}{x-3} \] 3. Apply the log rule: \[ \frac{1}{4} \ln|x-7| - \frac{1}{4} \ln|x-3| + c \] 4. Combine the logs: \[ \frac{1}{4} \log \left( \frac{x-7}{x-3} \right) + c \]

Step 4: Final Answer
The integral is $\frac{1}{4} \log \left( \frac{x-7}{x-3} \right) + c$.