【第九讲】一元函数积分学的计算
一、基本积分公式¶
牢记十组积分基本公式：
- \(\int x^k dx\) = \(\frac{1}{k+1}x^{k+1} + C,k\neq-1;\)
- \(\int \frac{1}{x^{2}}dx\) = \(-\frac{1}{x}+C\)
- \(\int \frac{1}{\sqrt{ x }}dx\) = \(2 \sqrt{ x }+ C\)
- \(\int \frac{1}{x} dx\) = \(\ln|x| + C\)
- \(\int e^x dx\) = \(e^x + C\)
- \(\int a^x dx\) = \(\frac{a^x}{\ln a} + C, a>0且a\neq 1\)
-
三角函数系列
- \(\int \sin x dx\) = \(-\cos x + C\)
- \(\int \cos xdx\) = \(\sin x+C\)
- \(\int \tan xdx\) = \(-\ln|\cos x| + C\)
- \(\int \cot xdx\) = \(\ln|\sin x|+C\)
- \(\int \frac{dx}{\cos x}\) = \(\int \sec xdx\) = \(\ln|\sec x+\tan x|+C\)
- \(\int \frac{dx}{\sin x}\) = \(\int \csc xdx\) = \(\ln|\csc x-\cot x|+C\)
- \(\int \sec ^{2}xdx\) = \(\tan x+C\)
- \(\int \csc ^{2}xdx\) = \(-\cot x+C\)
- \(\int\sec x \tan xdx\) = \(\sec x+C\)
- \(\int \csc x\cot xdx\) = \(-\csc x+C\)
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\[ \begin{cases} \int \frac{1}{1+x^{2}}dx = \arctan x + C, \\ \int \frac{1}{a^{2}+x^{2}}dx = \frac{1}{a} \arctan \frac{x}{a} + C(a>0) \\ \end{cases} \]
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\[ \begin{cases} \int \frac{1}{\sqrt{ 1-x^{2} }} dx = \arcsin x + C \\ \int \frac{1}{\sqrt{ a^{2}-x^{2} }} dx = \arcsin \frac{x}{a} + C(>0) \\ \end{cases} \]
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\[ \begin{cases} \int \frac{1}{\sqrt{ x^{2}+a^{2} }}dx = \ln(x+\sqrt{ x^{2}+a^{2} }) +C(常见a=1)\\ \int \frac{1}{x^{2}-a^{2}}dx = \ln|x+\sqrt{ x^{2}-a^{2} }|+C(|x|>a). \end{cases} \]
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\[ \int \frac{1}{x^{2}-a^{2}}dx = \frac{1}{2a} \ln \left|\frac{{x-a}}{x+a}\right| + C \quad \left( \int \frac{1}{a^{2}-x^{2}}dx= \frac{1}{2a} \ln\left| \frac{{x+a}}{x-a} \right| + C \right). \]
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\[ \int \sqrt{ a^{2}-x^{2} }dx = \frac{a^{2}}{2}\arcsin \frac{x}{a} + \frac{x}{2}\sqrt{ a^{2}-x^{2} } + C(a>|x|\geq 0). \]
- \(\int \sin ^{2}xdx\) = \(\frac{x}{2} - \frac {{\sin 2x}} {4} + C \quad \left( \sin ^{2}x= \frac{{1-\cos 2x}}{2} \right)\)
- \(\int \cos ^{2}xdx\) = \(\frac{x}{2} + \frac {{\sin 2x}} {4} + C \quad \left( \cos ^{2}x = \frac{{1 +\cos 2x}}{2} \right)\)
- \(\int \tan ^{2}xdx\) = \(\tan x-x+C\quad(\tan ^{2}x=\sec ^{2}x-1)\)
- \(\int \cot ^{2}xdx\) = \(-\cot x-x+C\quad(\cot ^{2}x=\csc ^{2}x-1)\)
二、不定积分的积分法¶
2.1 凑微分¶
常用凑微分公式：
- \(xdx\) = \(\frac{1}{2} d(x^{2})\)
- \(\sqrt{ x }dx\) = \(\frac{2}{3} d(x^{\frac{3}{2}})\)
- \(\frac{dx}{\sqrt{ x }}\) = \(2d(\sqrt{ x })\)
- \(\frac{dx}{x^{2}}\) = \(d\left( -\frac{1}{x} \right)\)
- 当 \(x>0\) 时，\(\frac{1}{x}dx\) = \(d(\ln x)\)
- \(e^xdx\) = \(de^x\)
- \(a^xdx\) = \(\frac{1}{\ln a}d(a^x),a>0,a\neq 1\)
- \(\sin xdx\) = \(d(-\cos x)\)
- \(\cos xdx\) = \(d(\sin x)\)
- \(\frac{dx}{\cos ^{2}x}\) = \(\sec ^{2}x dx\) = \(d(\tan x)\)
- \(\frac{dx}{\sin ^{2}x}\) = \(\csc ^{2}xdx\) = \(d(-\cot x)\)
- \(\frac{1}{1+x^{2}}dx\) = \(d(\arctan x)\)
- \(\frac{1}{\sqrt{ 1-x^{2} }}dx\) = \(d(\arcsin x)\)
2.2 换元法¶
基本思想：
\[
\int f(x)dx \xlongequal{x=g(u)} \int f[g(u)]d[g(u)] = \int f[g(u)]g'(u)du
\]
- 三角代换
- \(\sqrt{ a^{2}-x^{2} }\to\) \(令x=a\sin t,|t|< \frac{\pi}{2}\)
- \(\sqrt{ a^{2}+x^{2} }\to\) \(令x=a\tan t,|t|< \frac{\pi}{2}\)
- \(\sqrt{ x^{2}-a^{2} }\to\) \(令x=a\sec t,注意范围见P262\)
- 恒等变形后做三角函数代换，见 \(P 262\)
- 根式代换（见书本）
- 倒代换（见书本）
- 复杂函数的直接代换（见书本）
2.3 分部积分法¶
(1)分部积分公式：\(\int udv\) = \(uv-\int vdu\)
P 283 注 2 —— uv 的选取原则为：反对幂指三 \(\arcsin x, \ln x, x^{2}, e^x, \sin x\).
相对位置左边的宜选作 \(u\)，相对位置右边的宜选作 \(v\).
实际上这个相对位置顺序是和 \(u,v\) 在 \(\int udv\) 中的相对顺序保持一致的。
例 9.5 下注： 1. \(\int e^{ax}\sin bxdx = \\\frac{\left|\begin{matrix}(e^{ax})' & (\sin bx)' \\ e^{ax} & \sin bx\end{matrix}\right|}{a^{2}+b^{2}}+C\) 2. \(\int e^{ax}\cos bxdx = \\\frac{ \\\left|\begin{matrix}(e^{ax})' & (\cos bx)' \\e^{ax} & \cos bx \end{matrix}\right|}{a^{2}+b^{2}} \\+C\)
例如：
\[
\int e^{-x}\sin nxdx = \\
\frac{
\left|\begin{matrix}(e^{-x})' & (\sin nx)' \\
e^{-x} & \sin nx\end{matrix}\right|}
{(-1)^{2}+n^{2}} \\
= \frac{-e^{-x}\sin nx-ne^{-x}\cos nx}{(-1)^{2}+n^{2}}+C \\
\]
(2)分部积分的推广公式，见书本 P 284
2.4 有理函数的积分¶
略，见书 P 286
三、定积分的计算¶
-
区间再现公式： $$ \int_{a}^b f(x)dx = \int_{a}^b f(a+b-x) \, dx $$
-
华里士公式（点火公式🔥） $$ 🔥 = \begin{cases} \frac{{n-1}}{n}\cdot \frac{{n-3}}{n-2} \cdot ... \cdot \frac{2}{3} \cdot 1, n>1 且为奇数, \ \frac{{n-1}}{n}\cdot \frac{{n-3}}{n-2} \cdot ... \cdot \frac{1}{2} \cdot \frac{\pi}{2}, n>1且为偶数. \end{cases} $$
-
四组点火公式
- \(\int_{0}^{\frac{\pi}{2}} \sin^nxdx = \int_{0}^{\frac{\pi}{2}}\cos^nxdx = 🔥\)
- \(\int_{0}^{\pi}\sin^nxdx = 2\cdot 🔥\)
- \(\int_{0}^{\pi}\cos^nxdx = \begin{cases}0, n为正奇数 \\2\cdot🔥, n为正偶数. \\ \end{cases}\)
- \(\int_{0}^{2\pi}\sin^nxdx = \int_{0}^{2\pi}\cos^nxdx = \begin{cases} 0, n为正奇数, \\ 4\cdot🔥, n为正偶数. \\ \end{cases}\)
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例 9.8 结论（证明可利用区间再现构造方程，换元 \(u=\pi-t\) ）： $$ \int_{0}^\pi xf(\sin x)dx = \frac{\pi}{2} \int_{0}^\pi f(\sin x)dx. $$
四、变限积分的计算¶
4.1 求导公式¶
\[
F'(x) = \frac{d}{dx} \left[ \int_{\varphi_{1}(x)}^{\varphi_{2}(x)} f(t)dt \right]
= f[\varphi_{2}(x)]\varphi'_{2}(x) -
f[\varphi_{1}(x)]\varphi'_{1}(x)
\]
4.2 重要结论¶
- \(f(x)为可积的奇函数\implies \int_{0}^xf(t)dt为偶函数.\)
- \(f(x)为可积的偶函数\implies 只有\int_{0}^xf(t)dt为奇函数.\)
- \(f(x)是可积的以T为周期的周期函数\implies \int_{0}^xf(t)dt是以T为周期的周期函数.\)
五、反常积分的计算¶
- \(\Gamma函数\)
- 定义: \(\Gamma(\alpha) = \int_{0}^{\infty} x^{\alpha-1}e^{-x}dx \xlongequal{x=t^{2}} 2\int_{0}^{\infty} t^{2\alpha-1}e^{-t^{2}}dt\)
- 递推式：\(\Gamma(\alpha+1) = \alpha \Gamma(\alpha)\)
- 特殊值：\(\Gamma\left( \frac{1}{2} \right) = \sqrt{ \pi }\), \(\Gamma(1) = 1\)