From c6c53a0cb10bd801cc83790631e04ce03b8f5255 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 4 Jun 2021 23:50:50 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E4=B8=8D=E5=AE=9A=E7=A7=AF?= =?UTF-8?q?=E5=88=86=E5=88=86=E9=83=A8=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 85010 -> 103059 bytes ...efinite-integral-and-definite-integral.tex | 120 +++++++++++++++--- ...efinite-integral-and-definite-integral.pdf | Bin 166090 -> 166135 bytes ...efinite-integral-and-definite-integral.tex | 2 +- 4 files changed, 103 insertions(+), 19 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 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zmtjj@7?$Im5mOy`KWShXIa6cfz_5fq=N)XOri@HRCML`*#zv-Q%tow6T&6~*tR^fR zW~?Usy#GH>q#nkB5rQ*vrNYF6kpiq-{~uBry7?957)QL}s|yb4ZxNJrct4v;lG(}d zMDE1dA!5f^ohX%}(gPxrFWZVYwbUn_GU;qlFU{ndizZir6b zZ$&$_uMx|`2!zu(kR2G%7a0*8>{084LAKz)Ir3o`;uP(f-5Nus>|p2wY1e|YI8vMs z@uh`WGvPk$TJlD})PN!F`c?FP^?&f-T@IyNjvTJVkK0z{4aN6><+*~$ z9w=W9IlQ9P8=#@)GEM+wVcWCB^vtjP=N{I&(od+#fb^0$,所以正负号无误。 +若令$x=a\sin t$,则根据$\sin t\in(-1,1)$得到主区间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$。 -若令$x=a\cos t$,则根据$\cos t\in(-1,1)$,得到主区间:$t\in(0,\pi)$,从而代入式子$\sqrt{a^2-x^2}=a\sin t$,根据$t\in(0,\pi)$,所以$\sin t>0$,所以正负号无误。 +若令$x=a\cos t$,则根据$\cos t\in(-1,1)$,得到主区间:$t\in(0,\pi)$。 -所以这种情况不用考虑正负号。 - -\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt{1-x^2}}}$。 +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt{1-x^2}}}$。\medskip 令$x=\sin t$($t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$),所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$,$t=\arcsin x$。 +因为式子$\dfrac{1}{1+\sqrt{1-x^2}}>0$,单调递减,所以不用讨论正负号。 + $=\displaystyle{\int\dfrac{\cos t}{1+\cos t}\textrm{d}t=\int\dfrac{2\cos^2\dfrac{t}{2}-1}{2\cos^2\dfrac{t}{2}}\textrm{d}t=\int\textrm{d}t-\int\sec^2\dfrac{t}{2}\,\textrm{d}t=t-\tan}\dfrac{t}{2}+C$ $=t-\dfrac{\sin\dfrac{t}{2}}{\cos\dfrac{t}{2}}+C=t-\dfrac{\sin\dfrac{t}{2}\cos\dfrac{t}{2}}{\cos^2\dfrac{t}{2}}+C=t-\dfrac{\sin t}{1+\cos t}+C$ @@ -120,17 +116,37 @@ $=\arcsin x-\dfrac{x}{1+\sqrt{1-x^2}}+C$。 \paragraph{\texorpdfstring{$\sqrt{a^2+x^2}$:$x=a\tan t$}\ } \leavevmode \medskip -根据$\tan t\in R$,从而得到主空间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,代入$\sqrt{a^2+x^2}=a\sec t$,根据$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,所以$\cos t>0$,$\sec t>0$,所以正负号无误。 +根据$\tan t\in R$,从而得到主空间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$。 -所以这种情况不用考虑正负号。 +\textbf{例题:}求$\displaystyle{\int\dfrac{x^3+1}{(x^2+1)^2}\textrm{d}x}$。\medskip + +虽然本题目看着可以从分母解开平方,然后低阶分配,但是这分母是平方的式子很难分配,所以需要使用换元法。 + +令$x=\tan t$,$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,$x^2+1=\sec^2t$,$\textrm{d}x=\sec^2t\,\textrm{d}t$。 + +因为$(x^2+1)^2>0$,虽然$x^3+1$可能为负可能为正,但是都是单调递增的,所以不用考虑正负号。 + +$=\displaystyle{\int\dfrac{\tan^3t+1}{\sec^2t}\textrm{d}t=\int\dfrac{\sin^3t+\cos^3t}{\cos t}\textrm{d}t=\int\dfrac{\sin t(1-\cos^2t)+\cos^3t}{\cos t}\textrm{d}t}$ + +$=\displaystyle{\int\dfrac{\cos^2t-1}{\cos t}\textrm{d}(\cos t)+\int\dfrac{1+\cos2t}{2}\textrm{d}t}$ + +$=\displaystyle{\int\cos t\,\textrm{d}(\cos t)-\int\dfrac{1}{\cos t}\,\textrm{d}(\cos t)+\dfrac{1}{2}\int\textrm{d}t+\dfrac{1}{4}\int\cos2t\,\textrm{d}(2t)}$ + +$=\displaystyle{\int\cos^t-\ln\cos t+\dfrac{t}{2}+\dfrac{1}{4}\sin2t+C}$($\cos t$在$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$中为正) + +$\because\tan t=x$,$\therefore\sin t=\dfrac{x}{\sqrt{1+x^2}}$,$\cos t=\dfrac{1}{\sqrt{1+x^2}}$。 + +$=\dfrac{1+x}{2(1+x^2)}+\dfrac{1}{2}\ln(1+x^2)+\dfrac{1}{2}\textrm{arctan}\,x+C$。 \paragraph{\texorpdfstring{$\sqrt{x^2-a^2}$:$x=a\sec t$}\ } \leavevmode \medskip -根据$\sec t\in(-1,1)$,所以从而得到主空间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,代入$\sqrt{x^2-a^2}=a\tan t$,根据$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,所以$\tan t\in R$,所以此时不能保证转换后的$a\tan t>0$,此时必须对$x$分正负情况讨论。\medskip +根据$\sec t\in(-1,1)$,所以从而得到主空间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$。\medskip \textbf{例题:}求$\displaystyle{\int\dfrac{\sqrt{x^2-9}}{x}\textrm{d}x}$。 -令$x=3\sec t$。$\therefore\sqrt{x^2-9}=3\tan t$,$\textrm{d}x=3\sec t\tan t\,\textrm{d}t$。 +令$x=3\sec t$。$\therefore\sqrt{x^2-9}=3\tan t$,$\textrm{d}x=3\sec t\tan t\,\textrm{d}t$。\medskip + +因为式子$\dfrac{\sqrt{x^2-9}}{x}$的分子必然为为正,而对于分子在0两边的单调性不同,所以需要对$x$进行正负区分,又$x\in(-\infty,-3]\cup[3,+\infty)$,所以: 当$x>3$时,$\sec t>1$,即$t\in\left(0,\dfrac{\pi}{2}\right)$。 @@ -148,22 +164,78 @@ $=\sqrt{x^2-9}-3\arccos\dfrac{3}{-x}+3\pi+C$($3\arccos\dfrac{3}{x}=3\pi-3\arcc $=\sqrt{x^2-9}-3\arccos\dfrac{3}{-x}+C$。 +综上结果为$\sqrt{x^2-9}-3\arccos\dfrac{3}{\vert x\vert}+C$。 + \paragraph{辅助换元} \leavevmode \medskip 在使用换元法的时候有可能单个式子不能求出积分,而使用其他辅助式子加减在一起积分可以得到结果,从而能得到原式和辅助式子的积分结果。对于这类题目需要观察什么样的式子能让积分简单。 -\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{x+\sqrt{1-x^2}}}$。 +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{x+\sqrt{1-x^2}}}$。\medskip 令$x=\sin t$,所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 -$\therefore=\displaystyle{\int\dfrac{\cos t}{\sin t+\cos t}\textrm{d}t}$。 +$\because x+\sqrt{1-x^2}$可能为正可能为负,正负时单调性不同,所以令$ x+\sqrt{1-x^2}=0$,即$\sin t=-\dfrac{\sqrt{2}}{2}$,从而$t\in(-\dfrac{\pi}{2},-\dfrac{\pi}{4})\cup(-\dfrac{\pi}{4},\dfrac{\pi}{2})$。 + +$\therefore=\displaystyle{\int\dfrac{\cos t}{\sin t+\cos t}\textrm{d}t}$($t\in(-\dfrac{\pi}{4},\dfrac{\pi}{2})$)。 \medskip + +这时你会发现使用积化和差、万能公式、倍角公式都无法解出这个积分,所以这时候就需要另外一个辅助积分式子加上或减去这个式子,从而让和以及差更容易解出积分。这里根据式子特点让辅助式子分子为$\sin t$: + +令$I_1=\displaystyle{\int\dfrac{\cos t}{\sin t+\cos t}\textrm{d}t}$,$I_2=\displaystyle{\int\dfrac{\sin t}{\sin t+\cos t}\textrm{d}t}$。 + +$I_1+I_2=\displaystyle{\int\dfrac{\sin t+\cos t}{\sin t+\cos t}\textrm{d}t=\int\textrm{d}t=t}$。 + +$I_1-I_2=\displaystyle{\int\dfrac{\cos t-\sin t}{\sin t+\cos t}\textrm{d}t=\int\dfrac{\textrm{d}(\sin t+\cos t)}{\sin t+\cos t}}=\ln\vert\sin t+\cos t\vert +C$。 + +所以$I_1=\dfrac{1}{2}(\arcsin x+\ln\vert x+\sqrt{1-x^2}\vert)+C$。 + +同理$t\in(-\dfrac{\pi}{2},-\dfrac{\pi}{4})$也得到同样结果。 \subsection{分部积分} +因为分部积分法使用$\int u\,\textrm{d}v=uv-\int v\,\textrm{d}u$,所以基本上用于两项乘积形式的积分式子。 + \subsubsection{基本分部} +\paragraph{非幂函数优先} \leavevmode \medskip + +当幂函数与一些微分后能降低幂函数幂次的函数在一起时,先对非幂函数优先分部积分,结果与幂函数相乘可以消去幂次,以达到降低幂次的作用。 + +如$\int x^n\ln x\,\textrm{d}x$,$\int x^n\arctan x\,\textrm{d}x$,$\int x^n\arcsin x\,\textrm{d}x$。 + +\textbf{例题:}求$\int x^2\arctan x\,\textrm{d}x$。 + +$=\dfrac{1}{3}\int\arctan x\,\textrm{d}(x^3)=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\int x^3\,\textrm{d}(\arctan x)$ + +$=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\displaystyle{\int\dfrac{x^3}{1+x^2}\textrm{d}x}=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\displaystyle{\int\dfrac{x+x^3-x}{1+x^2}\textrm{d}x}$ + +$=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{3}\int x\,\textrm{d}x+\displaystyle{\dfrac{1}{6}\int\dfrac{\textrm{d}(1+x^2)}{1+x^2}}$ + +$=\dfrac{1}{3}x^3\arctan x-\dfrac{1}{6}x^2+\dfrac{1}{6}\ln(1+x^2)+C$。 + +\paragraph{幂函数优先} \leavevmode \medskip + +当幂函数与三角函数在一起微分时,因为三角函数无论如何积分都不会被消去,所以应该优先消去幂函数部分,从而降低幂次。 + +\textbf{例题:}求$\int x\tan^2x\,\textrm{d}x$。 + +$=\int x(\sec^2-1)\,\textrm{d}x=\int x\,\textrm{d}(\tan x)-\dfrac{x^2}{2}=x\tan x+\ln\vert\cos x\vert-\dfrac{x^2}{2}+C$。 + \subsubsection{多次分部} +\subsubsection{分部与换元} + +分部积分法和换元积分法经常一起使用。 + +\textbf{例题:}求$\int e^{\sqrt[3]{x}}\,\textrm{d}x$。 + +令$\sqrt[3]{x}=u$,从而$x=u^3$,$\textrm{d}x=3u^2\,\textrm{d}u$。 + +$=3\int e^uu^2\,\textrm{d}u=3\int u^2\,\textrm{d}(e^u)=3u^2e^u-3\int e^u\,\textrm{d}(u^2)=3u^2e^u-6\int e^uu\,\textrm{d}u$ + +$=3u^2e^u-6\int u\,\textrm{d}(e^u)=3u^2e^u-6ue^u+6\int e^u\,\textrm{d}u=3u^2e^u-6ue^u+6e^u+C$ + +$=3e^u(u^2-2u+2)+C=3e^{\sqrt[3]{x}}(x^{\frac{2}{3}}-2x^{\frac{1}{3}}+2)+C$。 + \subsection{有理积分} \subsubsection{高阶多项式分配} @@ -176,11 +248,23 @@ $=\displaystyle{\int\dfrac{x^3+9x-9x}{x^2+9}\,\textrm{d}x=\int\dfrac{x^3+9x}{x^2 $\displaystyle{=\int x\,\textrm{d}x-\dfrac{9}{2}\int\dfrac{\textrm{d}(x^2+9)}{x^2+9}}=\dfrac{x^2}{2}-\dfrac{9}{2}\ln(9+x^2)+C$。 -\subsubsection{低阶多项式分解} +\subsubsection{低阶多项式分配} -当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,则可以分解式子:$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x=\int\dfrac{f_1(x)}{g_1(x)}\,\textrm{d}x+\int\dfrac{f_2(x)}{g_2(x)}\,\textrm{d}x}$。 +当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,且$g(x)$不能因式分解为$g(x)=g_1(x)g_2(x)\cdots$时,则可以分解式子:$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x=a_1\int\dfrac{\textrm{d}(f_1(x))}{g_1(x)}+a_2\int\dfrac{\textrm{d}(f_2(x))}{g_2(x)}}+\cdots$,将积分式子组合成积分结果为分式的函数,如$\ln x$、$\arcsin x$、$\arctan x$等。 +\textbf{例题:}求$\displaystyle{\int\dfrac{x-1}{x^2+2x+3}\textrm{d}x}$。 +因为$x^2+2x+3$不能因式分解,所以考虑将分子按照分母形式进行分配。优先对高阶的$x$进行分配。 + +首先因为分子最高阶为$x$只比分母最高阶$x^2$低一阶,所以考虑将$x-1$分配到微分号内。 + +$\because\textrm{d}(x^2+2x+3)=2x+2$,而现在是$x-1$,所以: + +$=\displaystyle{\dfrac{1}{2}\int\dfrac{2x+2}{x^2+2x+3}\textrm{d}x-2\int\dfrac{1}{x^2+2x+3}\textrm{d}x}=\displaystyle{\dfrac{1}{2}\int\dfrac{\textrm{d}(x^2+2x+3)}{x^2+2x+3}}$ + +$-\displaystyle{\int\dfrac{1}{\left(\dfrac{x+1}{\sqrt{2}}\right)^2+1}\textrm{d}x}=\displaystyle{\dfrac{1}{2}\ln(x^2x+3)-\sqrt{2}\int\dfrac{\textrm{d}\left(\dfrac{x+1}{\sqrt{2}}\right)}{\left(\dfrac{x+1}{\sqrt{2}}\right)^2+1}}$ + +$=\dfrac{1}{2}\ln(x^2+2x+3)-\sqrt{2}\arctan\dfrac{x+1}{\sqrt{2}}+C$。 \section{定积分} diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index dedba8c2ff2f343df268545538d4b89fb836f3d7..a00fa5663d4956fce6dd349b0e021152ea3c06cd 100644 GIT binary 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a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -233,7 +233,7 @@ $\therefore\int\sec^3x\,\textrm{d}x =\dfrac{\sec x\tan x+\ln\vert\sec x+\tan x\v 假分式可以分解为多项式与真分式之和。 -真分式$\dfrac{P(x)}{Q(x)}$若可以分解为两个多项式的乘积:$\dfrac{P(x)}{Q(x)}=\dfrac{P_1(x)}{Q_1(x)}+\dfrac{P_2(x)}{Q_2(x)}$,则称为将真分式化为部分分式之和。 +真分式$\dfrac{P(x)}{Q(x)}$若可以分解为两个或多个多项式的乘积:$\dfrac{P(x)}{Q(x)}=\dfrac{P_1(x)}{Q_1(x)}+\dfrac{P_2(x)}{Q_2(x)}+\cdots$,则称为将真分式化为部分分式之和。 通过这种化简方式,可以在求以商的形式的有利函数的式子的积分时拆分因式,从而简化积分运算。这种简化运算主要体现在分数的积分为对数。