Q.35 The value of the integral
0.9
∫
0
dx / ((1 − x)(2 − x))
is ____________________.
Evaluating definite integrals involving rational functions is a common and important
topic in engineering mathematics and competitive exams like GATE.
In this article, we solve Q.25, which asks for the value of the definite integral:
∫00.9
dx⁄(1 − x)(2 − x)
We derive the correct answer step by step using partial fractions and explain
why the remaining options are incorrect.
Given Integral
∫00.9 dx⁄(1 − x)(2 − x)
Step 1: Partial Fraction Decomposition
1⁄(1 − x)(2 − x) = A⁄1 − x + B⁄2 − x
Solving:
1 = A(2 − x) + B(1 − x)
Comparing coefficients:
- A = 1
- B = −1
Therefore,
1⁄(1 − x)(2 − x) = 1⁄1 − x − 1⁄2 − x
Step 2: Integrate Term-Wise
∫ (1⁄1 − x−1⁄2 − x) dx = −ln|1 − x| + ln|2 − x|
Step 3: Apply Definite Limits (0 to 0.9)
At x = 0.9
−ln(0.1) + ln(1.1) = ln(11)
At x = 0
−ln(1) + ln(2) = ln(2)
Final Answer
ln(11) − ln(2) = ln 11⁄2
✅ Correct Answer:
ln11⁄2
MCQ Option Analysis
Option (A): ln(11/2) ✅
Correct. This is obtained by correct partial fraction decomposition
and exact application of limits.
Option (B): ln(11) ❌
Incorrect. Ignores subtraction of the lower limit term ln(2).
Option (C): ln(2) ❌
Incorrect. Represents only the lower-limit contribution.
Option (D): Integral diverges ❌
Incorrect. The integrand is continuous and finite on
[0, 0.9], so the integral converges.
Why This Question Is Important
- Tests partial fraction decomposition
- Reinforces definite integral evaluation
- Frequently asked in GATE, ESE, and university exams
- Builds strong fundamentals in rational integrals
Key Takeaway
- Always simplify integrands using partial fractions
- Apply upper and lower limits carefully
- Watch for logarithmic cancellations
