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Question #41854

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_{0}^{10} \left( x^2 [x+t] + 3x \{x+t\}^2 \right) dt$ , where $[ \cdot ]$ and $\{ \cdot \}$ denote the greatest integer and fractional part functions respectively. Find the value of $F'(2)$ .

Options

A.
310
B.
330
C.
370
D.
130

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

Algebra of Functions|Newton-Leibniz Formula with variable limits|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the Newton-Leibniz rule (differentiation under the integral sign).

A common trap for advanced students is blindly applying the Leibniz rule when the integrand contains moving jump discontinuities (like the floor function $[x+t]$ ).

Because the discontinuities depend on $x$ , their movement contributes Dirac delta boundary terms to the derivative.

Ignoring these terms yields an incorrect derivative (330).

The rigorous path requires evaluating the integral explicitly using properties of periodic functions and the identity $\int_{x}^{x+n} [u] du = nx + \frac{n(n-1)}{2}$ before differentiating.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Algebra of Functions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free