單擊此處編輯母版標(biāo)題樣式,單擊此處編輯母版文本樣式,第二級,第三級,第四級,第五級,*,*,P129,習(xí)題,5.2,1(1).6.9.,P133,習(xí)題,5.3,1(3)(6)(9).2(3)(5)(11).,3(3)(7)(9)(10).4(3)(8).,作 業(yè),預(yù)習(xí)：,P135141,12/11/2024,1,第十三講 不定積分（一）,一、原函數(shù)與不定積分概念,二、基本積分表,三、湊微分法,12/11/2024,2,一、原函數(shù)與不定積分概念,（,1,）,從運算與逆運算看,初等數(shù)學(xué)中加法與減法、乘法與除法、,乘方與開方等，都是互逆的運算。,微分運算是對一個可導(dǎo)函數(shù)求導(dǎo)數(shù)。,微分運算的逆運算是什麼？,問題：,這就是求原函數(shù)和不定積分的運算。,12/11/2024,3,（,2,）,從物理問題看,12/11/2024,4,（一）原函數(shù)的定義,12/11/2024,5,關(guān)于原函數(shù)有兩個理論問題,:,（,a,）,原函數(shù)的存在問題,結(jié)論,:,（,b,）,原函數(shù)的結(jié)構(gòu)問題,一個函數(shù)若存在一個原函數(shù)，,則它必有無窮多個原函數(shù)。,12/11/2024,6,定理,1,證,12/11/2024,7,12/11/2024,8,積分變量,積分常數(shù),積分號,（,二）不定積分的定義,記作,:,被積函數(shù),12/11/2024,9,積分曲線與積分曲線族,12/11/2024,10,例3,解,不是！,(1),12/11/2024,11,分段積分，得,12/11/2024,12,12/11/2024,13,12/11/2024,14,（三）不定積分的性質(zhì),（,1,）,不定積分與微分互為逆運算,12/11/2024,15,綜合,(3)(4),（,2,）,線性運算性質(zhì),12/11/2024,16,不定積分計算的基本思想：,求不定積分是求導(dǎo)的逆運算,導(dǎo)數(shù)基本公式,積分基本公式,微分法,積分法,反想,逆運算,怎樣計算不定積分？,12/11/2024,17,二、基本積分表,12/11/2024,18,12/11/2024,19,12/11/2024,20,12/11/2024,21,解,12/11/2024,22,解,12/11/2024,23,解,12/11/2024,24,12/11/2024,25,利用微分形式不變性,12/11/2024,26,三、湊微分法,定理,1,：（湊微分法）,證,利用微分形式不變性,12/11/2024,27,湊微分,怎樣應(yīng)用湊微分法？,12/11/2024,28,解,向哪個積分公式湊？,12/11/2024,29,解,12/11/2024,30,解,12/11/2024,31,解,12/11/2024,32,12/11/2024,33,解,12/11/2024,34,解,12/11/2024,35,解,12/11/2024,36,解,12/11/2024,37,解,12/11/2024,38,解,12/11/2024,39,常用微分公式,12/11/2024,40,12/11/2024,41,12/11/2024,42,