2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )

Transcription

1 Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) ISSN -X Online) ISSN 7-9 Prin) Reserch Aricle The clculion mehod for ind of definie inegrl Zhongui Zho College of Mhemics d Sisics, Norhes Peroleum Universiy, Dqing 68, Chin *Corresponding uhor Zhongui Zho Emil: zz69@6.com Absrc: This pper discussed d his ind of definie inegrl. Tn ) is n-h Qibiiofu polynomil, n,, L ) The conclusion is when n is even number,. And when n is odd number, ) ),, L ) Keywords: Definie inegrl;odd funcion, Mhemicl inducion; Qibiiofu series, error esimion.. When n is even number, he numericl vlue of d When n is even number, T n ) is even funcion, d rc is odd funcion, is even funcion, so he rc Tn inegrd of his inegrl is odd funcion. As he inegring rge [-,] is symmeric, so n,,, ) ). When n is odd number, he numericl vlue of d ) The relionship beween he definie inegrl, ) d ). If n is odd number,, hen rc T d rc ) ) d ) rc ) ) ) d ) d ) ) 8

2 Zho Z., Sch. J. Eng. Tech., ; A):8-6 So we c wor ou he follow equion. Becuse d lim d d lim rc ) rc ) When, hs no definiion) ) d d ) ) ) d d ) d d Though ), we c figure ou d ),d s d, d d ) so ) ) ) ) From he wo rigonomeric funcions s follows, ) ) ) ) ) ) We c figure ou ) ) ) Muliply he wo sides of his equion by, d ) ) ) clcule by inegrion. ) d ) ) ) d ) ) ) d 9

3 Zho Z., Sch. J. Eng. Tech., ; A):8-6 So Since ) ) d ) ) ) ) d ) ) d d ) d ) d ) From)we c figure ou 6 ) ),, ) 6) ) When n is odd number,the clculion formul of n Now we c prove he follow equion by inducion ) ),, From ) We c now h When, ) ) ) From ) We c now h When, 7) ) ) ) So when,,he equion 7) my be enble. If m, m, 7) is enble,so ) m m) m ) m ) m ) ; m ) ) is enble. 8) m m ) Form 6) we c wor ou h, when m, m m ) m) 6 m m) m ) m ) Form 8) we c figure ou by inducion hypohesis, m ) m ) 6 m ) 6 ) m m ) m ) ) m m m ) ) m ) m ) ) m ) m ) m ) m) m) m m m ) ) 6 ) m m ) ) ) m ) m) m ) ) m ) So he equion 7) c be enble., m ) 6

4 Zho Z., Sch. J. Eng. Tech., ; A):8-6 From ) d 6), we c figure ou, n n d ),, ) 9) ), n. Applicion By he definiion of Qibiiofu funcion series, he Qibiiofu series of rc ) is T T T rc Tn ) Among hem, d n,, ) From 9) we c figure our he vlue of n,, ) is ; n ), ), ) ),, ),. So he Qibiiofu series of rcg is ) ) T ) T ) T L ) T K L... And he Qibiiofu series of rc pril sum is ) P ) T ) T ) T L ) T K I is he bes me squre pproimion polynomil of rc. Ting P ) s pproimion of rc. ) rc ) T ) T L ) T K And ing P ) s pproimion of rc,clcule he pproimion of rc.. As T, T, T 6, so rc. ). )..) 8 8)6...) ). 76 ). 8 8) ) Now ry o me P ) which is he bes me squre pproimion polynomil of rc insed of rc error esimion. Since rc is coninuous funcion, d is firs-order derivive rc ) is bounded, so he Qibiiofu series of rc uniform convergence in rc. rc T T L T ) L ) By ), rc P T T ) L ) ) ) ) 6

5 Zho Z., Sch. J. Eng. Tech., ; A):8-6 As So By 7), we c figure ou By he formul C wor ou ) T ) ) T ) ) T n,,, ) n, rc P L ) ln ) ) ) ) ) ) -< ) ) ln ln ) ) ) Though ) d ), we c now rc P ln ) L ln ) ) ) ) ),, ) ) Now we use ) ) P ) insed of rc error esimion in [-,].Though ), we c figure ou 7 9 rc P ) ln ) ) ) ) 7 rc P7 ) ln ) ) ) ) ) 7 6 rc P9 ) ln ) ) ) ) ) ) 9 6 REFERENCES. Wg Renhong; Numericl Approimion. Higher Educion Press,999.. Li Qingyg; Numericl Anlysis. Huzhong Universiy of Science d Technology Press,.. Liu Fu; The clculion mehod of definie inegrl. Journl of Shi Norml Universiy Nurl Science Ediion), ; :-.. Lio Hui; A noe of definie inegrl wih rigonomeric funcion, Journl of Miyg Norml Universiy, ; 8:-6.. Du Shenggui; Inegrl mehod of rigonomeric funcion rionl formul, 99; :8-. 6