Definite Integration
Limits of Integral Families
Grade 12
Question:
<p>Let \(I_n = \int_{1/(n+1)}^{1/n} \tan^{-1}(nx) \, dx\), then \(\lim_{n \to \infty} n^2 I_n\) is equal to</p>
<p>(A) 1</p>
<p>(B) 0</p>
<p>(C) -1</p>
<p>(D) \(\frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Apply substitution and asymptotic analysis to evaluate the limit of a family of integrals.
<p>Using substitution $u = nx$ and analyzing the asymptotic behavior as $n \to \infty$, the integral can be evaluated to find $\lim_{n \to \infty} n^2 I_n = 1$.</p>
Correct Answer: A