삼각 치환, trigonometric substitution
적분의 테크닉으로 쓰임. (삼각치환적분, integration by trigonometric substitution, ITS)

$\displaystyle \sqrt{a^2-x^2}$
$\displaystyle x=a\sin\theta$

$\displaystyle \sqrt{a^2+x^2}$
$\displaystyle x=a\tan\theta$

$\displaystyle \sqrt{x^2-a^2}$
$\displaystyle x=a\sec\theta$

i.e. (Stewart: Table of Trigonometric Substitutions)

식	치환	θ의 범위	항등식
$\displaystyle \sqrt{a^2-x^2}$	$\displaystyle x=a\sin\theta$	$\displaystyle -\frac{\pi}2\le\theta\le\frac{\pi}2$	$\displaystyle 1-\sin^2\theta=\cos^2\theta$
$\displaystyle \sqrt{a^2+x^2}$	$\displaystyle x=a\tan\theta$	$\displaystyle -\frac{\pi}2<\theta<\frac{\pi}2$	$\displaystyle 1+\tan^2\theta=\sec^2\theta$
$\displaystyle \sqrt{x^2-a^2}$	$\displaystyle x=a\sec\theta$	$\displaystyle 0\le\theta<\frac{\pi}2\textrm{ or }\pi\le\theta<\frac{3\pi}2$	$\displaystyle \sec^2\theta-1=\tan^2\theta$

예 ¶

Q:
$\displaystyle \int\frac1{1+x^2}dx=\int dt$

Sol:
치환:

$\displaystyle t=\tan x$
$\displaystyle dt=\sec^2xdx$

식=

$\displaystyle \int\frac1{1+\tan^2t}\sec^2$

TBW

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