高等學(xué)十不定積分ppt課件

1、 第三章 不定積分一、原函數(shù)與不定積分 1原函數(shù)例2()2xx 也是 的原函數(shù)2x2x2(1)2xx21x 也是 的一個(gè)原函數(shù)2x2()2xcx2xc 叫做 的一個(gè)原函數(shù)2x(sin )cosxx sincosxx是的一個(gè)原函數(shù)(sin)cosxcxsin xc也是 的原函數(shù)cosx( )f x普通地假設(shè)( )( ),F xf x( )F xc是 函數(shù)的全體。 即( )f x ( )( )F xcf x證明 設(shè) 是 的一個(gè)原函數(shù)即( )x( )f x( )( )xf x由于( )( ),F xf x( )( )xf x( )( )xF x( )( )0 xF x ( )( )0 xF x( )

2、( )xF xc( )( )xF xc 的一切原函數(shù)都可以表示為( )F xc 求原函數(shù)的問(wèn)題就是不定積分問(wèn)題 (2)不定積分的定義 假設(shè)函數(shù) 是函數(shù) 的一個(gè)原函數(shù)，那么 函數(shù) 的原函數(shù)的全體 叫做函數(shù) 的不定積分。并記為( )F x( )f x( )f x( )F xc( )f x( )f x dx ( )F xc叫積分號(hào)x叫積分變量c叫積分常量( )f x dx叫做被積表達(dá)式( )f x叫做被函數(shù)例2xdx 2x dx 2xc313xcxe dx xecsin xdx cosxc二、原函數(shù)的幾何意義( )( )F xf x假設(shè)( )yF x 且求導(dǎo)數(shù): 知 求 ，也就是知一個(gè)函數(shù)的曲線求曲

3、線上每一點(diǎn)的切線斜率；求不定積分: 知 求 也就是知一個(gè)函數(shù)曲線上每一點(diǎn)的切線斜率，求這個(gè)函數(shù)。( )F x( )f x( )f x( )F x例 求經(jīng)過(guò)點(diǎn) 而它的斜率為 的曲線方程。(1,2)2x解 斜率為 的曲線方程2x2yxc且經(jīng)過(guò) 的方程為(1,2)21yx22xdxxc即yx0不定積分的幾何意義：就是一族積分曲線。 在積分曲線族上橫坐標(biāo)一樣點(diǎn)處的切線彼此平行三、不定積分的性質(zhì)證明1 不定積分的導(dǎo)數(shù)等于被積函數(shù)( )f x dx( )f x dx( )f xd( )f x dx( )f x dx ( )F xc ( )f x( )f x dxd( )f x dx( )f x dx dx

4、2 函數(shù)微分的不定積分等于該函數(shù)加上恣意常數(shù)( )F x dx( )F xc( )d F x( )F xc例(sin )x dxsin xccosxdx 3 被積函數(shù)中的常數(shù)因子，可以提到積分號(hào)外面。 ( )( )( )f xxx dx( )( )kf x dxkf x dx 4 幾個(gè)函數(shù)代數(shù)和的不定積分等于各個(gè)函數(shù)不定積分的代數(shù)和。( )( )( )f x dxx dxx dx(1)()kdxk dxkxc k四、不定積分公式表從不定積分的定義知：積分法是微分法的逆運(yùn)算，由導(dǎo)數(shù)公式可以直接推導(dǎo)出相應(yīng)的根本積分公式。是常數(shù)1(2)1nnxx dxcn1(3)lndxxcx21(4)1dxarc

5、tgxcx21(5)arcsin1dxxcx(6)cossinxdxxc(7)sincosxdxxc 221(8)seccosdxxdxtgxcx221(9)cscsindxxdxctgxcx (10)secsecxtgxdxxc(11)csccscxctgxdxxc (12)xxe dxec(13)lnxxaa dxca211(14)dxcxx 1(15)2dxxcx例41(1)dxx4x dx4 14 1xc 313cx 2(21)(3)xdxx2441xxdxx144xdxdxdxx224lnxxxc2(2)(5)x xdx5122(5)xxdx51225x dxx dx7227x321

6、073xxx xc32253xc 22(4)1xdxx 221 11xdxx 21(1)1dxx211dxdxxxarctgxc2(5)tg xdx22sincosxdxx221 coscosxdxx21cosdxdxxtgxxc(6)2xxe dx(2 )xe dx(2 )ln2xece221(5)(1)xxdxx x211()1dxxx2111dxdxxxln xarctgxc2(6)sin2xdx1 cos2xdx1(1 cos )2x dx1(sin )2xxc五、 第一類換元積分法湊微分法假設(shè)函數(shù) 的原函數(shù)是 ，且 ，那么 ( )f x( )F x( )ux ( )( )fxx dx

7、 ( )( )fx dx設(shè)( )ux( )f u du( )F uc ( )Fxc例(1)sin5xdx2(2)xe dx5d x (5 )x dx5dx155dxd x1sin555xd x1cos55xc (2 ) x dx2d x 2dx122dxd x2122xe d x212xec(3)tgxdxsincosxdxx1coscosdxx sincosxdxdx ln cosxc (4)ln sinctgxdxxc(5)2xdx12(2)xdx(2)d x(2)xdxdx(2)d xdx12(2)(2)xd x322(2)3xc10(6)()axbdx()d axb ()axb dxa

8、dx1()d axbdxa101()()axbd axba111()11axbca2(7)xxe dx2212xe dx22dxxdx212xdxdx212xec2(8)(1)xdx2(21)xxdx3213xxxc2(8)(1)xdx2(1)(1)xd x31(1)3xc10(9)(1)xdx10(1)(1)xd x111(1)11xc(10)sin2xdx1sin222xd x1cos22xc (10)sin2xdx2 sin cosxxdx2 sinsinxdx212sin2xc2sin xc(10)sin2xdx2 cos sinxxdx2 coscosxdx 212cos2xc 2c

9、os xc 23(11)xedx232()3xedx322332xec (12)sin()tdt1sin() ()tdt1cos()tc 2(13)2xxe dx22xe dx2xec221(14)dxxa221111()2xaa xaxa1112dxdxaxaxa1lnln2xaxaca1ln2xacaxa可計(jì)算出：2211ln2axdxcaxaax221(15)dxax2211 ( ) dxxaa2111 ( ) xdxaaa1xarctgcaa(16)secxdx1cosdxx2coscosxdxx21sin1 sindxx11 sinln21 sinxcx(17)cscxdx1sind

10、xx12sincos22dxxx212sin2cos2cos2xdxxx212cos22xdxxtg122xdtgxtgln2xtgc2xtgsin2cos2xx22sin22cossin22xxx1 cossinxxcscxctgxcscxdx ln2xtgcln cscxctgxc23(18)1xdxx 331131dxx13321(1)(1)3xd x11321 (1)1312xc 1322(1)3xc21(19)24dxxx21(21)3dxxx221(1)(1)( 3)d xx1133xarctgcsin(20)1 sinxdxxsin (1 sin )(1 sin )(1 sin

11、)xxdxxx22sinsin(1 sin)xxdxx2sincosxdxx21coscosdxx 222211coscoscoscoscosxdxdxdxxxx 1costgxxcx22sincosxdxx221 coscosxdxx1()dxd axba21()2xdxd x211()dxdxx1lndxdxx1(2)dxdxxxxe dxdecossinxdxdxsin( cos )xdxdx常用湊微分公式常用湊微分公式221sectancosdxxdxdxx221csc( cot )sindxxdxdxx21arctan1dxdxx2arcsin1dxdxx21 cos 2sin,2xx21cos 2cos2xx1sinsincoscos2 1coscoscoscos21sincossinsin2幾個(gè)常用的三角公式幾個(gè)常用的三角公式sin 2cos 3xx dx1sin 5sin2xx dx 1 1sin 55sin2 5x dxxdx11cos 5cos102xxC 211(1)xedxx263(2)331xdxxx24(3)23dxx 225(4)413xdxxx答案2xedx1(1)xedxx2xec263(2)331xdxxx221(331)331dxxxx2ln 331x