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Question #41553

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a strictly increasing continuous function defined as $f(x) = x^3 + \frac{7}{2}x + \int_0^x |t - 2| dt$ If $I = \int_{0}^{247} f^{-1}(x) dx$ , then the value of $3I$ is equal to:

Options

A.
3221
B.
3285
C.
1225
D.
5671

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

Inverse Function Integration|Monotonicity and Boundedness|Differentiability vs Continuity|Modulus Function Symmetry

Why This Question Is Tricky

Common Mistakes

This problem tests the integration of an inverse function where the original function contains a modulus-induced point of non-differentiability.

The standard trap is applying the fundamental theorem of calculus or integration by parts globally without partitioning the domain at the cusp ( $x=2$ ).

If a student ignores the modulus and assumes either branch of the integral $\int_0^x |t-2|dt$ applies globally over $[0,6]$ , they will coincidentally get the exact same incorrect integral value ( $\int_0^6 f(x)dx = 387$ because $\int_0^6 (x^2 - 4x)dx = 0$ ), leading to the highly attractive trap answer $3285$ .

This punishes rote application of IBP and forces rigorous piecewise analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Inverse Function Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free