From 1fab4192a6eb3c4580762bd292ec302370cbdfdc Mon Sep 17 00:00:00 2001
From: Elwood <3286545699@qq.com>
Date: Thu, 29 Jan 2026 17:20:31 +0800
Subject: [PATCH] vault backup: 2026-01-29 17:20:31

---
 .../高数素材/不定积分大集合——技巧篇.md   | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/素材/整合素材/高数素材/不定积分大集合——技巧篇.md b/素材/整合素材/高数素材/不定积分大集合——技巧篇.md
index b9d5036..6df5302 100644
--- a/素材/整合素材/高数素材/不定积分大集合——技巧篇.md
+++ b/素材/整合素材/高数素材/不定积分大集合——技巧篇.md
@@ -26,9 +26,10 @@ $$\int \frac{1}{(x^2+1)\sqrt{x^2-1}} dx$$
 $$\begin{align*}
 \int \frac{1}{(x^2+1)\sqrt{x^2-1}} dx &= \int \frac{\sec t \tan t}{(\sec^2 t+1)\tan t} dt \\
 &= \int \frac{\cos t}{\cos^2 t+1} dt \\
-&= \arctan(\sin t) + C \\
-&= \arctan\left(\frac{\sqrt{x^2-1}}{x}\right) + C
-\end{align*}$$
+
+&=  \frac{1}{2\sqrt{2}} \ln\left| \frac{\sqrt{x^2-1} + x\sqrt{2}}{\sqrt{x^2-1} - x\sqrt{2}} \right| + C\\
+\end{align*}\\
+$$
 【2.2】
 
 $$\int \frac{1}{x\sqrt{x^2-1}} dx$$