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Question #41445

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_n = \int_{-1}^{3} (x^n + n x^{n-1}) e^x \text{sgn}(x-1) \, dx$ for $n \in \mathbb{N}$ , where $\text{sgn}(\cdot)$ denotes the signum function. Find the value of $\sum_{n=1}^{5} \frac{I_n - (-1)^n e^{-1}}{3^n}$ .

Options

A.
$5e^3$
B.
$5e^3 - e + \frac{e}{243}$
C.
$5e^3 - 2e + \frac{2e}{243}$
D.
$5e^3 - e - \frac{e}{243}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Reduction Formulae|Method of Differences|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the application of integration by parts and reduction formulae across jump discontinuities.

A common trap is to blindly apply $\int u \, dv = uv - \int v \, du$ while ignoring the Dirac delta contribution from the derivative of the signum function.

This leads to the incorrect assumption that $I_n = f_n(3) + f_n(-1)$ , missing the $-2f_n(1)$ term generated at the boundary $x=1$ , which perfectly cascades into the trap answer $5e^3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Reduction Formulae
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free