第五章不定積分習(xí)題課參考答案

1、第五章不定積分習(xí)題課參考答案.pdf1第五章不定積分例1為的一個(gè)原函數(shù)，則＝。xexcos?)(xf)(xf?解:由已知為的導(dǎo)函數(shù)，即)(xfxexcos?()(cos)sinxxfxexex?????所以，()(sin)cosxxfxexex??????例2設(shè)為的原函數(shù)，求：。)(xfxx?2??10)(dxxfx解:由已知為的導(dǎo)函數(shù)，即xx?2)(xf2()xxfx???所以，1112324300011()d()d()d43xfxx

2、xxxxxxxxxC???????????例3求下列不定積分：①；②??dxxx3)1(???dxxx22121分析：用分項(xiàng)積分法，解:①；5311332222212(1)331dd33dxxxxxxxxxxxxx???????????????????故753132222(1)26d2275xxxxxxCx???????②222221232(1)3dd2d3arctan2111xxxxxxxCxxx???????????????????

3、??例4計(jì)算不定積分：。??dxxx)12(1解:1(21)21211ddddd(21)(21)(21)2121xxxxxxxxxxxxxxx?????????????????????????故1dlnln21(21)xxxCxx??????例5設(shè)，則＝。???CxFdxxf)()(?xdxxfcos)(sin第五章不定積分習(xí)題課參考答案.pdf3解:11dd(ln)lnlnlnlnxxxCxxx?????例10求下列不定積分：①；②；

4、③；④??12xxdx??224xxdx???dxex1???dxxxx2)ln(ln1分析：①令或，②令，③令，④令txsec?tx1?txtan2?xet???1tx1?第二類換元法常用代換有：根式代換，三角代換，倒代換，其中三角代換可使被積函數(shù)消去根號而有理化，尤為多用使用第二類換元法求出原函數(shù)后一定要將變量回代常用第二類換元法積分類型：①，令；②，令；??dxbaxxfn)(nbaxt????dxxaxf)(22taxsin?③

5、，令；④，令；??dxxaxf)(22taxtan???dxaxxf)(22taxsec?⑤倒代換，常用于消去被積函數(shù)分母中的因子tx1?x解:①；2222d1d111d()arcsin11111()xxCxxxxxxx????????????????法２當(dāng)時(shí)，0x?12222dd11darcsinarcsin1111xtxttttCCtxxxtt??????????????????????令當(dāng)時(shí)，同理。0x?②2tan()222222