Using Substitution

Substitution rewrites the variable to simplify the integrand.

定积分的换元法

换元法要点

1. Change limits: map $a,b$ to new limits $\alpha,\beta$.
2. Monotone & continuous: pick $\varphi$ monotone/differentiable so $f(\varphi(t))\varphi'(t)$ is continuous.
3. Pattern spotting: try trig/hyperbolic for roots like $\sqrt{a^2-x^2}$; linear for $(ax+b)^k$.
4. Combine with parts: sometimes substitute to isolate a factor, then integrate by parts.

常见换元策略

• Linear: $x=at+b$ (shift/scale).
• Power: $x=t^{\,k}$ or $t=x^{1/k}$.
• Trig: $x=a\sin t$, $x=a\tan t$ for radicals $\sqrt{a^2-x^2}$, $\sqrt{x^2+a^2}$.
• Exponential/log: $x=e^t$, $t=\ln x$.

应用例子

例 1 $\int_0^1 x e^{x^2} dx = \tfrac12(e-1)$ via $u=x^2$.
例 2 $\int_0^1 \dfrac{1}{\sqrt{1-x^2}} dx = \int_0^{\pi/2}1\,dt=\tfrac{\pi}{2}$ using $x=\sin t$.
例 3 $\int_1^{e^2} \dfrac{\ln x}{\sqrt{x}}dx = 4$ (substitution + parts as in steps).

练习题

练习 1

用换元法计算 $\int_0^1 \dfrac{x}{\sqrt{1-x^2}} dx$。

练习 2

用换元+分部积分求 $\int_1^{e^2} \dfrac{\ln x}{\sqrt{x}}\,dx$。

练习 3

改编自2022考研数学一填空题

设 $t = \tan \theta$，计算 $\int_0^{\pi/4} \dfrac{1}{1+\tan^2 \theta}\,d\theta$。

Summary

本文出现的符号

中英对照