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Question #40657

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $m \in [-50, 10]$ such that the definite integral $I(m) = \int_{m}^{-41-m} \frac{39}{x^2+41x+40} \ln\left( -\frac{x+1}{x+40} \right) dx$ evaluates to a real number equal to zero. Find the sum of all elements in $S$ .

Options

A.
-779
B.
-1220
C.
-820
D.
-441

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Expansion of Products of Binomials|V-N Method (Partial Fractions)|Definite Integral Properties (King's Property)|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to combine algebraic decomposition (V-N method/partial fractions) with definite integral properties, while strictly enforcing domain constraints.

The primary trap lies in blindly applying King's Property ( $x \to a+b-x$ ) to conclude the integral is always zero, leading students to sum all integers in the given range $[-50, 10]$ .

However, the natural logarithm requires its argument to be strictly positive, restricting the valid domain of integration to $(-40, -1)$ .

Furthermore, the integral diverges at the boundaries, meaning the limits must strictly reside within this interval.

Students must reject extraneous solutions outside this domain to find the correct sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Expansion of Products of Binomials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free