Problem Sets

Published: October 23, 2025

Last Update: 2025-10-14

Remark:

Per the sequence of the course outline, this set does not yet cover problems related to exponential functions, inverse functions, and logarithms—these will be included in subsequent sets.
根据课程大纲的内容顺序安排，此部分练习暂未涵盖与指数函数、反函数及对数函数相关的题目 —— 这类题目将在后续练习中纳入。

(\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x})
$\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x}$
[\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x}]
$\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x}$
$\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x}$
$\lim_{x \to 0} \dfrac{\int_{\cos x}^{1} e^{-t^{2}} dt}{\sin^{2} x}$
$\int \sec^{3} x \, dx$
$\int_{0}^{\pi} \sqrt{\cos^{2} x - \cos^{4} x} \, dx$
$\int \frac{1}{1 + \sqrt[3]{x + 2}} dx$
$\int x\sin^{-1} x\,dx$
Suppose that $ x $ and $ y $ satisfy the following equation: $ x = \int_{0}^{y} \frac{1}{\sqrt{1+4t^{2}}} dt$. Show that $ \frac{d^2y}{dx^2} $ is proportional to $ y $ and find the constant of proportionality.

\(\int x\sin^{-1} x\,dx\)