From 076b1852fce85327c47760ce41656bff6761c17a Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sun, 27 Jun 2021 23:14:02 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=A7=AF=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 137099 -> 141692 bytes ...efinite-integral-and-definite-integral.tex | 34 ++++++++++++++++-- ...efinite-integral-and-definite-integral.pdf | Bin 180468 -> 182538 bytes ...efinite-integral-and-definite-integral.tex | 25 ++++++++++--- 4 files changed, 51 insertions(+), 8 deletions(-) diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 415727a1c830119d78d4c39cf02089d2099d754f..793812eff48a89dec096319bee424945c727e84a 100644 GIT binary patch delta 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a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex index 9214141..893feb5 100644 --- a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -95,9 +95,21 @@ $=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\se 需要利用到有理积分的高阶多项式分配与低阶多项式因式分解。 +\paragraph{反三角换元} \leavevmode \medskip + +当积分式子中存在$\arcsin x$、$\arccos x$、$\arctan x$这种反三角函数时,可以考虑将其令为$t$来进行简化计算。从而$x$分别为$\sin t$、$\cos t$、$\tan t$。 + +\paragraph{指对换元} \leavevmode \medskip + +当积分式子存在指数函数$e^x$或对数函数$\ln x$时,可以考虑令其为$t$,从而$x$分别为$\ln t$和$e^t$。 + +\paragraph{倒数换元} \leavevmode \medskip + +当被积函数的分母的幂次要比分子高两次以及以上时,令$x=\dfrac{1}{t}$。 + \paragraph{有理换元} \leavevmode \medskip -书上这个类型属于有理函数部分,我这里移动到第一类换元中。即将无理因式设为一个变量,从而提高式子的阶数,消除无理式变为有理式。 +书上这个类型属于有理函数部分,我这里移动到第一类换元中。即将无理因式直接设为一个变量,从而提高式子的阶数,消除无理式变为有理式。 有理换元时无理因式中的$x$必须是一阶的,如$\sqrt[3]{x+6}=u$,若是二阶需要利用第二类换元(三角换元),否则则无法消去无理因式项,因为$x$不能用单个的$u$来表示,如$\sqrt{x^3+6}=u$,$u=\sqrt[3]{u^2-6}$。 @@ -243,12 +255,20 @@ $I_1-I_2=\displaystyle{\int\dfrac{\cos t-\sin t}{\sin t+\cos t}\textrm{d}t=\int\ 同理$t\in(-\dfrac{\pi}{2},-\dfrac{\pi}{4})$也得到同样结果。 +\paragraph{\texorpdfstring{$\sqrt{ax^2+bx+c}$}\ } \leavevmode \medskip + +当被积函数含有根式$\sqrt{ax^2+bx+c}$时,需要对其配方变成$\sqrt{\varphi^2(x)+k^2}$、$\sqrt{\varphi^2(x)-k^2}$、$\sqrt{k^2-\varphi^2(x)}$三种形式再进行三角函数换元。 + \subsection{分部积分} 因为分部积分法使用$\int u\,\textrm{d}v=uv-\int v\,\textrm{d}u$,所以基本上用于两项乘积形式的积分式子。 +函数积分难度为:反三角函数、对数函数、幂函数、指数函数、三角函数。越往右求导越难,左边更应该当$u$进行求导,而右边更适合做$v$进行积分。 + \subsubsection{基本分部} +如果不是多次分部就是基本分部,目的都是为了降低积分式子幂次。 + \paragraph{非幂函数优先} \leavevmode \medskip 当幂函数与一些微分后能降低幂函数幂次的函数在一起时,先对非幂函数优先分部积分,结果与幂函数相乘可以消去幂次,以达到降低幂次的作用。 @@ -277,7 +297,7 @@ $=\int x(\sec^2-1)\,\textrm{d}x=\int x\,\textrm{d}(\tan x)-\dfrac{x^2}{2}=x\tan \subsubsection{多次分部} -对于一部分通过微分形式不会发生变化的函数,所以需要多次积分,然后利用等式求出目标值。 +对于一部分通过微分形式不会发生变化的函数,所以需要多次积分,然后利用等式求出目标值。即三角函数和指数函数,这两种积分形式不变,指数函数一次积分保持不变,而三角函数两次积分保持不变。 如:$\int e^x\sin x\,\textrm{d}x$,$\int e^x\cos x\,\textrm{d}x$。 @@ -307,6 +327,14 @@ $=\sin^2x\cdot e^x-\sin2x\cdot e^x+2e^x\cos2x+4\sin2x\cdot e^x-8\int e^x\cdot\co $=\dfrac{e^x(5\sin^2x-\sin2x+2\cos2x)}{5}+C=e^x\left(\dfrac{1}{2}-\dfrac{1}{5}\sin2x-\dfrac{1}{10}\cos2x\right)+C$ +$\int uv'''\,\textrm{d}x=\int u\,\textrm{d}(v'')=uv''-\int v''u'\,\textrm{d}x$ + +$\int u'v''\,\textrm{d}x=\int u'\,\textrm{d}(v')=u'v'-\int v'u''\,\textrm{d}x$ + +$\int u''v'\,\textrm{d}x=\int u''\,\textrm{d}v=u''v-\int vu'''\,\textrm{d}x$ + +$\therefore\int uv'''\,\textrm{d}x=uv''-u'v'+u''v-\int u'''v\,\textrm{d}x$。 + \subsubsection{分部与换元} 分部积分法和换元积分法经常一起使用。 @@ -479,7 +507,7 @@ $\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}=\lim\limits_{n\to\inf 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zLAJ~)gErT+eZv}`jrh9v38@dKqz<}nhQa~L^VsL~!|vW(Hk0Yz+buO|5uJG$2rHz&xwk)XX#O~wR;cYr^72rsp8 z_ruZ$a9t%=vnHq3K9p9ep3hXl*Z0aea$eED1D$;4pqaiW@lagqy7g0pjJynmxjx~S zvijf&(l2RJ==>Nt(jrc>`<<%nY)#x+B2hmdM@xu2CZ zdp605O_hb)#L2V1Ol8_-H(#dYh?IInwWE=1d02Y~nqfdciS{-jc-!ikv0UuLK#N`G%ovv%gf$bN%&3rNEG-jw{TLfZ zcC?;Yew?r!N0NXnN1E7P4t}&DfW=uLwiDK5-}5pq{J-)_U0MHAHrwgeO7VwyJQz&=r67V4^3|)V2;Y za6sOZjBQ)0JrZ<{WY~A=E3*KcbY||;(+-HchP7byM&}#L61xiH9Qtz^o0Bc*U*-Ik zFt&6j@2k-%?H%<(qw7@ua^V(PvK*OXy=_neW7N1?O@w9QqPS3lJ83;n#?+`@0OY6L z1Qda-dofE)n=4^T%@0=2R3~1JE8Jgf0kkTO@~vvE8iEf9HDraW^&M*77J6j`>1xI< zehZ>qSqtm&n!WQp*PollZn)1YBs*EiSxxHdRa$,做定积分$\int_a^tf(x)\,\textrm{d}x$,对这种变上限积分的极限$\lim\limits_{t\to+\infty}\int_a^tf(x)\,\textrm{d}x$就是$f(x)$在无穷区间$[a,+\infty)$上的反常积分,记为$\int_a^{+\infty}f(x)\,\textrm{d}x$。 @@ -476,9 +487,11 @@ $\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)(a<\xi b)$。 无穷限反常积分\textcolor{violet}{\textbf{定义:}}$\int_{-\infty}^{+\infty}f(x)\,\textrm{d}x=\int_{-\infty}^0f(x)\,\textrm{d}x+\int_0^{+\infty}f(x)\,\textrm{d}x$。 +对于无穷区间的反常积分要求的就是$\lim\limits_{x\to\infty}f(x)$。 + \subsubsection{无界函数} -若$f(x)$在点$a$的任意一个邻域内都无界,则$a$就是$f(x)$的瑕点(无界间断点),无界函数的反常积分又称为瑕积分。 +若$f(x)$在点$a$的任意一个邻域内都无界,则$a$就是$f(x)$的瑕点(无穷间断点),无界函数的反常积分又称为瑕积分。 设$f(x)$在区间$(a,b]$上连续,点$a$为$f(x)$的瑕点,任取$t>a$,作定积分$\int_t^bf(x)\,\textrm{d}x$,则对变下限的定积分求极限的$\lim\limits_{t\to a^+}\int_t^bf(x)\,\textrm{d}x$就是函数$f(x)$在区间$(a,b]$上的反常积分,记为$\int_a^bf(x)\,\textrm{d}x$。 @@ -488,6 +501,8 @@ $\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)(a<\xi b)$。 若$f(x)$在区间$[a,c)\cup(c,b]$上连续,$c$为瑕点,则$\int_a^bf(x)\,\textrm{d}x=\int_a^cf(x)\,\textrm{d}x+\int_c^bf(x)\,\textrm{d}x$。 +对于无界函数的反常积分要求的就是$\lim\limits_{x\to a}f(x)$。 + \subsection{* 反常积分的判敛} \subsection{不定积分与定积分的区别与联系}