∫ 三角函数的积分
sin、cos、tan 等
📐 基本积分公式
| \(f(x)\) | \(\int f(x) \, dx\) |
|---|---|
| \(\sin x\) | \(-\cos x + C\) |
| \(\cos x\) | \(\sin x + C\) |
| \(\frac{1}{\cos^2 x}\) | \(\tan x + C\) |
| \(\frac{1}{\sin^2 x}\) | \(-\cot x + C\) |
| \(\tan x\) | \(-\ln|\cos x| + C\) |
| \(\cot x\) | \(\ln|\sin x| + C\) |
💡 记忆要点:
• sin 的积分 → 负 cos
• cos 的积分 → sin(没有负号)
⚡ 含线性表达式 (ax + b)
规则:除以 x 的系数!
| \(\int \sin(ax+b) \, dx\) | \(-\frac{1}{a}\cos(ax+b) + C\) |
| \(\int \cos(ax+b) \, dx\) | \(\frac{1}{a}\sin(ax+b) + C\) |
| \(\int \frac{1}{\cos^2(ax+b)} \, dx\) | \(\frac{1}{a}\tan(ax+b) + C\) |
例题:
\(\int \sin(3x) \, dx = -\frac{1}{3}\cos(3x) + C\)
\(\int \cos(2x+1) \, dx = \frac{1}{2}\sin(2x+1) + C\)
\(\int \frac{1}{\cos^2(5x)} \, dx = \frac{1}{5}\tan(5x) + C\)
✏️ 例 1:基本积分
计算:\(\int (3\sin x - 2\cos x) \, dx\)
解答:
\(\int 3\sin x \, dx - \int 2\cos x \, dx\)
\(= 3 \cdot (-\cos x) - 2 \cdot \sin x + C\)
\(= -3\cos x - 2\sin x + C\)
答案:\(-3\cos x - 2\sin x + C\)
✏️ 例 2:定积分
计算:\(\int_0^{\frac{\pi}{2}} \cos x \, dx\)
解答:
\(\Big[ \sin x \Big]_0^{\frac{\pi}{2}}\)
\(= \sin\frac{\pi}{2} - \sin 0\)
\(= 1 - 0 = 1\)
答案:1
📚 积分中常用的三角恒等式
基本恒等式
\(\sin^2 x + \cos^2 x = 1\)
二倍角公式(非常重要!)
\(\sin^2 x = \frac{1 - \cos(2x)}{2}\)
\(\cos^2 x = \frac{1 + \cos(2x)}{2}\)
积化和差
\(\sin x \cos x = \frac{1}{2}\sin(2x)\)
🔢 sin²x 和 cos²x 的积分
\(\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C\)
\(\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C\)
💡 记忆要点:唯一区别在于 sin(2x) 的符号!
✏️ 例 3:sin²x 的积分
计算:\(\int \sin^2 x \, dx\)
解答:
步骤 1:使用恒等式
\(\sin^2 x = \frac{1 - \cos(2x)}{2}\)
步骤 2:代入积分
\(\int \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 - \cos(2x)) \, dx\)
步骤 3:求解
\(= \frac{1}{2} \left( x - \frac{\sin(2x)}{2} \right) + C\)
\(= \frac{x}{2} - \frac{\sin(2x)}{4} + C\)
答案:\(\frac{x}{2} - \frac{\sin(2x)}{4} + C\)
🔄 换元积分
当三角函数的幂出现连同其导数时:
类型 1:\(\int \sin^n x \cos x \, dx\)
代换:\(u = \sin x\),\(du = \cos x \, dx\)
结果:\(\frac{\sin^{n+1} x}{n+1} + C\)
类型 2:\(\int \cos^n x \sin x \, dx\)
代换:\(u = \cos x\),\(du = -\sin x \, dx\)
结果:\(-\frac{\cos^{n+1} x}{n+1} + C\)
类型 3:\(\int \frac{\sin x}{\cos^n x} \, dx\)
代换:\(u = \cos x\)
结果:\(\frac{1}{(n-1)\cos^{n-1} x} + C\)
✏️ 例 4:sin³x·cos x
计算:\(\int \sin^3 x \cos x \, dx\)
解答:
代换:\(u = \sin x\)
\(du = \cos x \, dx\)
积分变为:
\(\int u^3 \, du = \frac{u^4}{4} + C\)
回代到 x:
\(= \frac{\sin^4 x}{4} + C\)
答案:\(\frac{\sin^4 x}{4} + C\)
✏️ 例 5:cos⁴x·sin x
计算:\(\int \cos^4 x \sin x \, dx\)
解答:
代换:\(u = \cos x\)
\(du = -\sin x \, dx\) → \(\sin x \, dx = -du\)
积分变为:
\(\int u^4 \cdot (-du) = -\frac{u^5}{5} + C\)
回代到 x:
\(= -\frac{\cos^5 x}{5} + C\)
答案:\(-\frac{\cos^5 x}{5} + C\)
✏️ 例 6:tan x
计算:\(\int \tan x \, dx\)
解答:
改写:
\(\int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx\)
代换:\(u = \cos x\),\(du = -\sin x \, dx\)
\(= \int \frac{-du}{u} = -\ln|u| + C\)
回代到 x:
\(= -\ln|\cos x| + C\)
答案:\(-\ln|\cos x| + C\)
✏️ 例 7:sin x · cos x
计算:\(\int \sin x \cos x \, dx\)
解答(两种方法):
方法 1:恒等式
\(\sin x \cos x = \frac{1}{2}\sin(2x)\)
\(\int \frac{1}{2}\sin(2x) \, dx = -\frac{1}{4}\cos(2x) + C\)
方法 2:换元(\(u = \sin x\))
\(\int u \, du = \frac{u^2}{2} + C = \frac{\sin^2 x}{2} + C\)
答案:\(-\frac{\cos(2x)}{4} + C\) 或 \(\frac{\sin^2 x}{2} + C\)
💡 两个答案等价!(差一个常数)
✏️ 例 8:含换元的定积分
计算:\(\int_0^{\frac{\pi}{2}} \sin^2 x \cos x \, dx\)
解答:
代换:\(u = \sin x\),\(du = \cos x \, dx\)
更换积分限:
当 \(x = 0\):\(u = \sin 0 = 0\)
当 \(x = \frac{\pi}{2}\):\(u = \sin\frac{\pi}{2} = 1\)
积分:
\(\int_0^1 u^2 \, du = \Big[ \frac{u^3}{3} \Big]_0^1 = \frac{1}{3} - 0 = \frac{1}{3}\)
答案:\(\frac{1}{3}\)
📋 总结表
| 积分 | 结果 |
|---|---|
| \(\int \sin(ax) \, dx\) | \(-\frac{1}{a}\cos(ax) + C\) |
| \(\int \cos(ax) \, dx\) | \(\frac{1}{a}\sin(ax) + C\) |
| \(\int \sin^2 x \, dx\) | \(\frac{x}{2} - \frac{\sin(2x)}{4} + C\) |
| \(\int \cos^2 x \, dx\) | \(\frac{x}{2} + \frac{\sin(2x)}{4} + C\) |
| \(\int \tan x \, dx\) | \(-\ln|\cos x| + C\) |
| \(\int \sin^n x \cos x \, dx\) | \(\frac{\sin^{n+1} x}{n+1} + C\) |
| \(\int \cos^n x \sin x \, dx\) | \(-\frac{\cos^{n+1} x}{n+1} + C\) |
💡 考试提示
1️⃣ sin 带负号
sin 的积分得到负 cos
2️⃣ 除以系数
\(\sin(ax)\) → 除以 a
3️⃣ sin² 和 cos²
使用二倍角公式
4️⃣ 幂 × 导数
\(\sin^n x \cdot \cos x\) → 代换 \(u = \sin x\)
📝 总结
\(\int \sin x \, dx = -\cos x + C\)
\(\int \cos x \, dx = \sin x + C\)
记忆:括号内有系数 → 除以它
幂带导数 → 换元