《7.2 Integration by Parts.pdf》由会员分享，可在线阅读，更多相关《7.2 Integration by Parts.pdf（18页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、 Integration by Parts x xe dx * Characteristics: The product of different functions. * Thinking idea: The derivation rule of product. lnxxdx arcsin xdx Integration by Parts has continuous derivatives. Then, we can obtain: Let functions ( ) and ( )uu xvv x ()uvu vuv()uvuvu v Integral: on both sides u
2、v dx ()uv dx u vdx udv uvvdu Integration by parts udvuvvdu The general principle of choosing (1) (2) Integration by Parts Integration by Parts andudv vis easy to obtain; vdu is more easier than .udv cos ,ux Find Method1 Example 1 cosxxdx udvuvvdu 2 , 2 x v sin,duxdx ,dvxdx Then, 2 coscos() 2 x xxdxx
3、d 2 cos 2 x x 2 sin 2 x xdx Obviously, choosing, u dvis very important, Otherwise, the integration becomes more difficult. Find cosxxdx ,ux Method2 Example 1 udvuvvdu sinvx ,dudxcos,dvxdx cossinxxdxxdx sinxxsinxdx sincosxxxC ,uxLet Find Example 2 x xe dx xx xeeC udvuvvdu x ve,dudx x dve dx xx xe dxx
4、de xx xee dx 2 ,ux Let Find Example 3 2x x e dx 22xx x e dxx de 2x x e 2 2() xxx x exeeC udvuvvdu x ve2,duxdx x dve dx 2 x xe dx （by parts again） Find Example 4 3 lnxxdx 4 1 ln 4 xx 4 1 ln 4 x dx 44 11 ln 416 xxxC udvuvvdu 4 3 lnln 4 x xxdxxd ln ,ux 4 , 4 x v 1 ,dudx x 3 dvx dx 11 Find Example 5 sin
5、 x exdx sin x xde sin(sincos ) 2 x x e exdxxxC sin x ex(sin ) x e dx sincos xx exexdx sincos xx exxde sin x ex(coscos ) xx exe dx (sincos )sin xx exxexdx Pay attention to the circular form u udv uudv Summary () Integration by Parts:udvuvvdu ( )sin,( )cos,( ), kx nnn P xaxdxP xaxdxP x e dx , are cons
6、tants,k a( ) is an -order polynomial. n P xn uuu Summary () ( )arcsin,( )arctan, nn P xxdxP xxdx ( )ln n P xxdx uu u We know, using the following two formulas arcsinarccos, 2 xx arctanarccot, 2 xx These integrations can be transformed into: ( )arccos, n P xxdx ( )arccot n P xxdx ( )arcsin, n P xxdx
7、( )arctan n P xxdx Summary () sin(),cos(), kxkx eaxb dxeaxb dx u should be the same type function every time. , are constant numbers.k a We can select u and dv by random arcsinarcsinxxxdx Questions and Answers arcsin xdx 2 1 arcsin 1 xxxdx x 2 arcsin1xxxC udvuvvdu Questions and Answers 2 6 1 lnttdt 2 7 1 1 ln 7 tdt 2 727 1 1 111 (ln ) 77 tttdt t 7 72 1 11 (2 ln2) 77 7 t 128127 ln2 749 udvuvvdu Questions and Answers cos x exdx cos x xde cos(cos ) xx exe dx cossin xx exxde cos(sin(sin ) xxx exxee dx (sincos )cos xx exxexdx cos(sincos ) 2 x x e exdxxxC Integration by Parts