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MTH 102 MTH 102 Past Questions Objective Question
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Question

By using Partial fraction technique, ∫ dx/(x²-x) is
Options
A <code>-ln(x²-x)</code>
B <code>ln|(x-1)/x|</code>
Correct Answer
C <code>ln|x/(x-1)|</code>
D <code>ln(x-1)</code>
Correct Answer

Option B is the correct answer.

Detailed Explanation

To integrate ∫ dx/(x²-x) using partial fractions:
  1. Factor the denominator:x²-x = x(x-1)
  2. Set up the partial fraction decomposition:1/(x(x-1)) = A/x + B/(x-1)
  3. Multiply by the common denominator x(x-1):1 = A(x-1) + Bx
  4. Solve for A and B:
    • Let x = 0: 1 = A(0-1) + B(0) => 1 = -A => A = -1.
    • Let x = 1: 1 = A(1-1) + B(1) => 1 = B.
  5. Substitute A and B back into the decomposition:1/(x(x-1)) = -1/x + 1/(x-1)
  6. Integrate each term:∫ (-1/x + 1/(x-1)) dx = ∫ (-1/x) dx + ∫ (1/(x-1)) dx= -ln|x| + ln|x-1| + C
  7. Use logarithm properties to combine the terms:= ln|(x-1)/x| + C
This matches option B.

Hint

Factor the denominator first. Then use partial fraction decomposition to break down the integrand into simpler fractions that are easier to integrate. Remember the properties of logarithms to simplify the final answer.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 432 of 930
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