Integrals of Irrational Functions

Irrational integrals often use substitutions that remove radicals.

代换法

例子

例 1 $\int \dfrac{1}{\sqrt{x}} dx$
Let $u=\sqrt{x}$, $x=u^2$, $dx=2u\,du$ → result $2\sqrt{x}+C$.

例 2 $\int \dfrac{1}{x\sqrt{x^2 - 1}} dx$
Let $x=\sec t$, $dx=\sec t \tan t\,dt$, $\sqrt{x^2-1}=\tan t$ → result $t+C = \arccos \tfrac{1}{x}+C$.

Summary

本文出现的符号

中英对照