第二換元積分二、第二類換PDFcreatedwithpdfFactorytrialversion x1x2dxdxcostdt dxcostdt1sin2x1x2dx 1sin2sintcos2PDFcreatedwithpdfFactorytrialversion

定理

設(shè)x(t)是單調(diào)的、可導(dǎo)的函數(shù)f(x)dxf[(t)](t)dtt (f(x)dxf[(t)](t)dtt (其中1(x)是x(t)的反函數(shù).即：在求積分f(x)d時(shí)可以作變量代換：xPDFcreatedwithpdfFactorytrialversion 設(shè)(t)為f[(t)](t)的原函數(shù)F(x[1(x)]Fxddtf[(t)](t

1(t)f[(t)]f Fx)為fx)的原函數(shù) f(x)dxF(x)C[1(x)]f(x)dxf[(t)](t)dtt (PDFcreatedwithpdfFactorytrialversion 所求積分f變量代換：x(tf[(t)]d(t)f[(t)](tf[(t)](t)dtF(t)f(x)dxF[1(x)]PDFcreatedwithpdfFactorytrialversion 例 求

a2x2 (a 令xasin

dxacostdtt,aa2sin2aa2sin2 a

x2

acosa2cos2

a2

(1cosa2

(t

1sin2t)C.PDFcreatedwithpdfFactorytrialversion a2

(tsintcost)Cax1(xaa21(xa

]C 例

4x2 令x2sin dx2cos

t, 2PDFcreatedwithpdfFactorytrialversion

4x2dx2sint

2cos44sin232sin3tcos2tdt32sint(1cos44sin22t32(cos2tcos4t)dcos2t

cost cost) 4x244x2314x25C4x2 PDFcreatedwithpdfFactorytrialversion 例

(ax2解令xatantdxasec2

t, 2 dx

asec2x2a2

asecx2tasec lnx2tax2a2xx2a2

C1.x2a2 x2a2PDFcreatedwithpdfFactorytrialversion 例 求

(ax2a2 (1)當(dāng)xa令xasectdxasecttan

t0, dxasecttan

2x2a2secx2x2

atanCxtx2aalnsecCxtx2aaCPDFcreatedwithpdfFactorytrialversion (2當(dāng)xa令x

dx

(t1xadxt1at2t2

x

x2x2x x2

lnx

x2x2PDFcreatedwithpdfFactorytrialversion 1x1xadxaxx

a當(dāng)xa時(shí)，令xasect也可以(t2計(jì)算過程與xa

,在整個(gè)定義域上，令xa則t(0,。tan2t

tan

tan 一樣需要討論P(yáng)DFcreatedwithpdfFactorytrialversion a2a2x2xasina2a2xatanx2x2xasectax2bx通過配方，可以化為上述三類型之一。PDFcreatedwithpdfFactorytrialversion 說(shuō)明化掉根式，三角代換并不是唯一的方 11令令t 1x2

t42t2t42t2x5x1x21x2

2dx

1t52t3tC1(84x2

3x4

CPDFcreatedwithpdfFactorytrialversion 1x1x11

dx2dx12

x4u1xuu1x

d(1x2uu1x21 2

u

1

2u32 u u1x21(84x231x2

CPDFcreatedwithpdfFactorytrialversion 例

1e

t 1t 1e

ext2xlnt2 dx

t22t22t2

1e

t

tttt

C2ln1ex1xCPDFcreatedwithpdfFactorytrialversion 當(dāng)被積函數(shù)含有根式時(shí)，可令整個(gè)根式為

axb或

axbcx例 求

1 x 令t x xt21,dx111112(tln1t)1

1

11

ln1

)PDFcreatedwithpdfFactorytrialversion 例

1 令t

11xx

t2

,dx

原式

1t2

t2

(t2t

1t(t

1)22[1t212[1t21t2

)dt

2arctantttt

t t 1t1111x1111PDFcreatedwithpdfFactorytrialversion 例 求

x(1 x1t22 令xt1t22

(t dx

6t x(1 x

t3(1t2)t21

1t

61

1t

6[tarctant]C

xarctan

x]CPDFcreatedwithpdfFactorytrialversion 說(shuō)明當(dāng)分母的階較高時(shí)可采用倒變換x1t例 求

x(x7 令x1dx1 t t12t7 t12t7x(x7

17t t

2

t21ln|12t7|C1ln|2x7|1ln|x|C PDFcreatedwithpdfFactorytrialversion 例 求

t,(1x2

2令xtan 1x2sec2t,dxsec令xtan1(1x2

dx

(1tan2

sec21sec2dtcos2 (1cos2t)dt1[dt cos2td(2t 11[t1sin2tC]1[arctanx1

PDFcreatedwithpdfFactorytrialversion 例 求

1e

令e t即xlntdx111 111ex(1 )dt1dt

d(t t lntln(t1)C

tee PDFcreatedwithpdfFactorytrialversion tanxdxlncosxcotxdxlnsinxsecxdxlnsecxtanxcscxdxlncscxcotx dx1arctanxC;a2x2 PDFcreatedwithpdfFactorytrialversion dx1lnx

x2 xa2 dxarcsina2ax2x2 x2x2

PDFcreatedwithpdfFactorytrialversion 第一換元法：處理復(fù)合函數(shù)（湊微分f[(x)](x)dx[f(u)du]u(x)第二換元法：主要處f(x)dxf[(t)](t)dtt (熟記：基本積分表PDFcreatedwithpdfFactorytrialversion 1111x1arcsinx

x1 1x

x1x12x

1 11x

4

x1x(x1)24xx(x1)24x x3

)]PDFcreatedwithpdfFactorytrialversion sinxcosxdx1 1arctan(sin2x)2tan3xsec

lntanx13

xsecx

cosxsin1(lntanx)22PDFcreatedwithpdfFactorytrialversion PDFcreatedwithpdfFactorytrialversion 例 求

x x2

t1x2令x t1x2令x dx

11t1t1 t ut

t1t

dt2

t1t

dtPDFcreatedwithpdfFactorytrialversion 111 du111112

d(111211111u3 131x211x23

C11x2PDFcr