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Question #40647

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_{0}^{x^2} \frac{1}{\sqrt{279 - 25t}} dt$ . The function $F'(x)$ is expanded as an infinite Maclaurin series $\sum_{n=0}^{\infty} a_n x^{2n+1}$ . Let $R$ be the radius of convergence of this series, and $k = [R]$ be the greatest integer less than or equal to $R$ . Find the value of $\int_{0}^{k} \frac{1}{x^2 - 6x + 25} dx$ .

Options

A.
$\frac{1}{4} \arctan\left(\frac{3}{4}\right)$
B.
$\frac{1}{4} \arctan\left(\frac{4}{3}\right)$
C.
$\frac{1}{5} \arctan\left(\frac{3}{5}\right)$
D.
$\frac{1}{3} \arctan\left(\frac{3}{4}\right)$

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

Binomial Theorem for Negative/Fractional Indices|Leibniz Rule for Differentiation under Integral|Greatest Integer Function (GIF)|Completing the Square in Quadratics

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of advanced calculus and series expansion.

The primary trap lies in correctly identifying the radius of convergence for a non-standard binomial expansion after applying the Leibniz rule.

Students often miscalculate $\sqrt{279}/5$ or fail to apply the Greatest Integer Function correctly, leading to incorrect integration limits.

Furthermore, the integral requires completing the square, where arithmetic fatigue from the constants 25 and 279 can induce sign errors, specifically missing the negative sign in the lower limit evaluation of the arctangent function.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Theorem for Negative/Fractional Indices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free