From 0438abfc4bb164c2976d5a99f06fab25e176f722 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 3 Jun 2021 23:52:32 +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=E4=B9=A0=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 65958 -> 85010 bytes ...efinite-integral-and-definite-integral.tex | 68 ++++++++++++++++-- advanced-math/knowledge/0-perpare/perpare.pdf | Bin 206736 -> 295967 bytes advanced-math/knowledge/0-perpare/perpare.tex | 68 +++++++++++++++++- 4 files changed, 129 insertions(+), 7 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 36491a92424cf9c84ae15ba36572d6668c5ee784..16707e27d3995caa8aa785d73c4935f86e17ad04 100644 GIT binary patch delta 78222 zcmZs?Q*@?Lw=^2twr$&X(y?uJ?7XpUqhs5)ZQHiffB!h+>~HLSw=SNm8uOW}YE@N9 z5LjY5I5N4Cm;^mD13L`)(cSeM3@dj6G8urGg@a2#0LIzH$;`+O#$)4JTR!fP1Frj3 zF=tKsC{EKY!MMc)Oc+lT#i~bA zxk$~2-WK?yUc8Uw&5{gb8MFIdp*L^iwzKl@VXq)Rrr?(`EZ8A`$HBXjrUa^ZWGO%h zB1`A_*1^BLrG*Ci3OKmyXV>5_;Pd`nLodvwf%s*uYEU?>{~%%=YTpXGj~}S;hv2Cd zf~eaZ_&40%^Tz7*_o0)ZU|zCw2s}jfq4_T`=ID+JbejV!wW@^bBt1@A&n6{J{IjiJ zu44@uCBhnENw+giLnH`hz+m8mKwqdQT{ zPZid&{h{zR0`4#SR6GueU?fX;ZsiPI?i;aKf~NPQJ5$}djzE7$`@R?*;9$T|{w0^l ziW<%VJ5O3?EF(7Gy|GBN?>%a$C;E~WkQa=WzzIeb@{&t+yR=E9+#f$!-&lyM+qqX| zC)Ycr%Z*rsN5z0Wxw&^=e0A*G5L8ma1py)(Ch-L&By%SGxDVJ*6h*v?a_ zfo(NfRC~prRL4d`#psrq6mfU!1PaI7VqI)Uv+B)rO+10G`!$-oW+uEQryp5h2vQ_f zB_&}^A8K2;k;(Uw(}UK7Z|T8wmclHL`MUxiTPvuLl@{Qu1i*kca%Ui1=OU30-aI>2 zVyOA)FG8Sv{PYQa{amTGVUkY3n9AMj5WQDfgQ>a%(~m(F&Ycedlbb9mnG`#lRiUDkFZ zgzC7)x5;WP#$IKonMv+v9^UIpy&H}XBBGV#Wh~czq#pdnRCsxfg>wEj0q7wzIB_ zV;F_4lRyxXUhX@;P-CVS2;~8U&^bz|DG>))+?Zt z2>AkTK+HjNKq&67JFXXSUxb!AFkMG6UN-av5MUiUsuU zY@V%FysmI(vP?&EQJp|m_Bj5CQ1%fmM%~*2{xPmNMcx)!Gr?auLkFY-)mMcTQ9405 zw!{;oO)@sx#EPn+u&ouzIK}1-xJ{{1`Lv@L_$Tn~P3rX^x7N^yJDg{bug1TuvvYzQULo^JS_ zi;3ZBJ5tZ(#d!LOU{44huFGO_wnDUHq?CT0mao&jMf?FgPx7be-R3n}n!I!+9#!3-s}XfhYl z`?G#UY2_m4kE-%RSS{ZiZlQzhQLpvZqRm#3hDJo|QAvrZkH@{DiGnDdk9S#m3fu8glXU17~%18 zn(_k5ed)hL&d!=-Bux(BV&(j=m{04-#*uL&_dL>GtkJ7RkPax8w;~d0Mr=lM_x~=N zs1nIEL!edfn>~f)-94_7bf1N));@P!N8rsh2=4AQaBE_Hi*pNb?=Ph&(fdTl7d*Hn z{N9ag1f<>G3SRrYmev){F>!Z(x=66H^^e;twg8My-SawnyRi=6o^SJhqU-GiQhk%; zhD1E$n?;tenXlxMITYUxOA~_ahP-CTj73x3y(wdujf(tx+ltuakvEIjka!{rW*!5T z2o&X=7~*ydV{)#!*%Q4`YRJ8<6(zKH6${?c9OcibnFx6<6FFmb1ywObRZ!OuH<^FJ zO#oyjbUizVahxl0=IKM%^V$xxro4=$b3D#>p_^|qKf_{qEYCsNawt{40+(x!+58_H zHB?SX##pW7j%(_6yjM8L*btvhZI{fL zF;3@C`{(3M>4Bsv_3zyyN6kUh?X6Hf*u*r_K0$U^%JoZrUk=kpsm9H)(7@`K3I6VU z^3J&;Ln>KNw+oki#HOtTt>{3}D?sa45Vi!}kW9t>d-i-g%bh3Vh(LWgHEmE{tocVC z`8K*8kw(vnkAQi#f4dof$asvGw+^AGS2d2li=(`zT2;DiU@L)-Y^_4l4*ii-2TH9i zpZ(g~Vj9nD$bNz0#&}QK=gv)uYo9m5e))vg5Ff(*!ml?J+fzT1oPGl4ZouH2#0!?C z(&bDsrB2;k)JW)-iY+#L3|+_|53x>ttcn}MNcrN7V1leJ20x*e@9&iBabXL{mY>cU z1ZysYo7b?@jyt-6w==d1g#Jwm^LH_D5@~!9rf@a;i%jldGq5|Ny=Bs(HL|SKh3+*v z2@#sm8e1$V(8P=L&5RSw;GL<#?g zLO<1D7(ArKkabW|9WdOss1ZqOD(I39blU)9;L{tp7B}8Ch&HX#vLTuWw>^Lu>DW@^ zFktGbZmZjvj^^g3Cj9Rugaf!p*fl*@C)_B%4RBFgTnWIpCO}_}3lB(`YG=i`T{7x8 z6@ykD^VF8CARBu72v9f^6MAEDl#@vpMM{6@e7JWeA?_Ep1~h4_`~K=3(!z!dL{!;h zf%EF@d&%(-PDfz4Fl}ehctER9Aiz4N#)!za=gsu=6qY$flKLCmH#td#AES&QHpsjt zCd_Dt}^~Hm&!@UW|@SFIziY5H@&c$LtpP zKd-r>bN0*^0`1*l%r^fxX;~RELP-c{bf5+71fu}oQ||1mnCVGsmCJ$=aXPELf2ke^ zhf$1hIv+eb1Lzr`$YdN~;IfKw)r^I%qZ=I7*ZJwzqho8tB=yBcqqfM(YjfbMSXVh_ z#{a@QIDqNd4h()6LwqHXYOl3pkpU9e_?gF^y3C=1Q`gFE%ui(#7fr95>~W7nMhu;k z;A=fN-ScE}0#D3+yY^!aVGMu#a9Uc9o6IJ0zaF_F0pCnT^)fCOQiC-%sC^B#PQhVQ zSE|`>CuZ31`3=#0nyDG!o4_gwHlo*k{lZy`q5)}~ptXDrn@><|dysl7Q&NYi2`n=+ z1RD%Ea%38oB*Y(gOU+hAi>6nzzq{;|XPnuMOtU|9o?ph$1bIJBTaDT2(Rr#*B0&+I zg?EW?@)ZBY}A{Ea4I{B(ut*b*+Xw*K}1#AQNkhdqsMlLPPHwESeod3{v)Z_RD z#5Y-Lr3MI_&SNNCN;v)Q1z~N1F-Z{~gJQdWhY7Ic*DR@#hvb-%506zR0LEsQ)0Dvz z@|q9P$4$4PL6zOMWPafH3PM_~!&tVEyNk>!n+Q%h6rm_qYvo40IGyYu&8)N8xQ@QV^a4BWU08a!G20pHQBZpIR?2dX2#pCi=n_qQIf3b{Rtz4% z4~)Q*`=BW%#SBxJb}`rXl35OwHm7CM0&MP;Bgqe3EvLa9hl+Q*KIldULhhMNd@v8= zT6czpROi9=#*RC&HheHyx0(oMYQ0l21voSw`hT0Ym4-43ykSA>G;DVk@ zf78VTWlyC0I7Bcz2`Hb}*T`|^oR4x*n}q*4Uv<>8m($2>=X^O_Tl8`LG|sTI2Rxb9 zQ94YZq|TTzp`T9@>#h0SdPrv22 ziyxh3U^iTYKGxQy-33C!_)%6KeZ%S)DcT}Xxxt=iwB0N{+5l?2-f}cDG2rP%s<$9m zR!YQBxd@S~XcGPcE5fXNR(Y*oDqS6+ZQ{%;h7L_vrk% zejspYK5rr&T8dFNVG1fCEv3Y@PCn^&20`3?`#I}aAvPpD%FflMBO!?1TZ};cD|`m7 z$6~dPN|ik&ZwVRW)7}5!7a)~jZwT^27q-&BA4-+J*~?NBg@pPcR9IF+q^BuoBT7tP z5c_F4aKFJ6bJDkHCtwFq__ixM2hwH~4|TP#kAor}4CGZst+m@2qB#;Fz%kHx%*gQ6 z)vQ7kD!g{%TG+;Ju9QXF{w#yk(A!aC{ybj0qSrrZDN4VrMT#M&44^>=&&WmDs`=D! zpIDimUy@n?9!pUv&T&kP+dZ5HvBSPq#i zhRe3FB(Zn5S&QEDFR8?T1Qg?ruqh0>L<2VKQ% z%0)eR0HCDHf1h*y1s$^5^8OzP_5al{%-r1nl~6D1{{MuURCQ$kPKC`Es9$Ca;{
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b159536..2532b04 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 @@ -60,6 +60,10 @@ \subsection{换元积分} +使用换元法做了换元之后是要带回式子中的,也就是说要保证反函数的存在才能代入有意义。为了保证反函数的存在,因此要保证原函数的单调性,所以要有一个规定的范围来使原函数保证单调。 + +对于第二类换元特别要考虑这方面,而第一类换元一般不考虑。 + \subsubsection{第一类换元} \paragraph{聚集因式} \leavevmode \medskip @@ -94,11 +98,65 @@ $=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\se \subsubsection{第二类换元} -\begin{enumerate} - \item $\sqrt{a^2-x^2}$:$x=a\sin t(a\cos t)$。 - \item $\sqrt{a^2+x^2}$:$x=a\tan t$。 - \item $\sqrt{x^2-a^2}$:$x=a\sec t$。 -\end{enumerate} +当使用第二类换元时需要考虑$x$定义域与换元后式子的正负号的问题,因为根号一定为正,而在定义域上的不同部分,换元式子正负号可能不同。 + +\paragraph{\texorpdfstring{$\sqrt{a^2-x^2}$:$x=a\sin t(a\cos t)$}\ } \leavevmode \medskip + +若令$x=a\sin t$,则根据$\sin t\in(-1,1)$得到主区间:$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,从而代入式子$\sqrt{a^2-x^2}=a\cos t$,根据$t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$,所以$\cos t>0$,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:$t\in(0,\pi)$,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。 + +所以这种情况不用考虑正负号。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt{1-x^2}}}$。 + +令$x=\sin t$($t\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$),所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$,$t=\arcsin x$。 + +$=\displaystyle{\int\dfrac{\cos t}{1+\cos t}\textrm{d}t=\int\dfrac{2\cos^2\dfrac{t}{2}-1}{2\cos^2\dfrac{t}{2}}\textrm{d}t=\int\textrm{d}t-\int\sec^2\dfrac{t}{2}\,\textrm{d}t=t-\tan}\dfrac{t}{2}+C$ + +$=t-\dfrac{\sin\dfrac{t}{2}}{\cos\dfrac{t}{2}}+C=t-\dfrac{\sin\dfrac{t}{2}\cos\dfrac{t}{2}}{\cos^2\dfrac{t}{2}}+C=t-\dfrac{\sin t}{1+\cos t}+C$ + +$=\arcsin x-\dfrac{x}{1+\sqrt{1-x^2}}+C$。 + +\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。 + +所以这种情况不用考虑正负号。 + +\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 + +\textbf{例题:}求$\displaystyle{\int\dfrac{\sqrt{x^2-9}}{x}\textrm{d}x}$。 + +令$x=3\sec t$。$\therefore\sqrt{x^2-9}=3\tan t$,$\textrm{d}x=3\sec t\tan t\,\textrm{d}t$。 + +当$x>3$时,$\sec t>1$,即$t\in\left(0,\dfrac{\pi}{2}\right)$。 + +$=\displaystyle{\int3\tan^2t\,\textrm{d}t=3\int(\sec^2t-1)\textrm{d}t}$ + +$=3\tan t-3t+C=\sqrt{x^2-9}-3\arccos\dfrac{3}{x}+C$。 + +当$x<-3$时,$\sec t<-1$,即$t\in\left(-\dfrac{\pi}{2},0\right)$。 + +$=\displaystyle{-\int3\tan^2t\,\textrm{d}t=-3\int(\sec^2t-1)\textrm{d}t}$ + +$=-3\tan t+3t+C=\sqrt{x^2-9}+3\arccos\dfrac{3}{x}+C$($\tan t<0$) + +$=\sqrt{x^2-9}-3\arccos\dfrac{3}{-x}+3\pi+C$($3\arccos\dfrac{3}{x}=3\pi-3\arccos-\dfrac{3}{x}$) + +$=\sqrt{x^2-9}-3\arccos\dfrac{3}{-x}+C$。 + +\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$\displaystyle{\int\dfrac{\textrm{d}x}{x+\sqrt{1-x^2}}}$。 + +令$x=\sin t$,所以$\sqrt{1-x^2}=\cos t$,$\textrm{d}x=\cos t\,\textrm{d}t$。 + +$\therefore=\displaystyle{\int\dfrac{\cos t}{\sin t+\cos t}\textrm{d}t}$。 \subsection{分部积分} diff --git a/advanced-math/knowledge/0-perpare/perpare.pdf b/advanced-math/knowledge/0-perpare/perpare.pdf index b711390bf65e0abb9fa12ae57dc9af99e05c2c5a..88abd2be20283490296d5844ef8bd12e1704f3cf 100644 GIT binary patch literal 295967 zcma&NLy#~$w6)u|ZQHhO+qP}nwr$(CZS!s0-S@kgoVx$uOj4<&Y7dgVc2+$}NEJlH 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