Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s Theoem Chii-Huei Yu * Depatment of Management and Infomation Nan Jeon Univesity of Science and Technology Tainan City Taiwan *Coesponding autho: chiihuei@nju.edu.tw Received Decembe 9 4; Revised Mach 4; Accepted Mach 3 4 Abstact This aticle takes advantage of the mathematical softwae Maple fo the auiliay tool to study si types of definite integals. The infinite seies foms of these definite integals can be obtained by using Paseval s theoem. In addition we popose some eamples to do calculation pactically. The eseach methods adopted in this study involved finding solutions though manual calculations and veifying these solutions using Maple. Keywods: definite integals infinite seies foms Paseval s theoem Maple Cite This Aticle: Chii-Huei Yu Solving Some Definite Integals Using Paseval s Theoem. Ameican Jounal of Numeical Analysis vol. no. (4): 6-64. doi:.69/ajna---5.. Intoduction In calculus and engineeing mathematics couses we leant many methods to solve the integal poblems including change of vaiables method integation by pats method patial factions method tigonometic substitution method and so on. In this pape we study the following si types of definite integals which ae not easy to obtain thei answes using the methods mentioned above. π sinh( cos )cosh( cos ) π sin( sin )cos( sin ) π π sinh( cos )cosh( cos ) sinh ( cos ) π sin( sin )cos( sin ) π () () (3) (4) (5) (6) whee is a eal numbe. We can obtain the infinite seies foms of these definite integals by using Paseval s theoem; these ae the majo esults of this pape (i.e. Theoems and Theoems ). The study of elated integal poblems can efe to [-6]. On the othe hand we povide some definite integals to do calculation pactically. The eseach methods adopted in this study involved finding solutions though manual calculations and veifying these solutions by using Maple. This type of eseach method not only allows the discovey of calculation eos but also helps modify the oiginal diections of thinking fom manual and Maple calculations. Fo this eason Maple povides insights and guidance egading poblem-solving methods.. Main Results Fistly we intoduce a notation and a definition and some fomulas used in this aticle... Notation Let z = a + ib be a comple numbe whee i = a b ae eal numbes. We denote a the eal pat of z by Re(z ) and b the imaginay pat of z by Im(z )... Definition Suppose f () is a continuous function defined on [ π ] the Fouie seies epansion of f () is a + ( ak cosk + bk sin k) k = whee π π a = f ( ) and a = π k f ( )cosk π π b k = π f ( )sin k fo all positive integes k.
Ameican Jounal of Numeical Analysis 6.3. Fomulas Poof.3.. Eule s Fomula e i = cos + i sin whee is any eal numbe..3.. DeMoive s Fomula n (cos + i sin ) = cosn + i sin n whee n is any intege and is any eal numbe..3.3. ([7]) sinh( p + iq) = sinh pcosq + i cosh psin q p q whee ae eal numbes. (y Fomulas.3.3 and.3.4).3.4. ([7]) cosh( p + iq) = cosh pcosq + isinh psin q p q whee ae eal numbes. And.3.5. Taylo Seies Epansion of Hypebolic Tangent Function ([8]) tanh( z ) = n n ( ) (n)! n z n whee z π is a comple numbe z < and n ae enoulli numbes fo all positive integes n..3.6. Taylo Seies Epansion of Hypebolic Cotangent Function ([8]) n n coth( z ) = + n z whee z is a z (n)! comple numbe < z < π. Net we intoduce an impotant theoem used in this study..4. Paseval s Theoem ([9]) If f () is a continuous function defined on [ π ] and f ( ) = f (π ). Suppose the Fouie seies epansion of f () is a + ( an cos n + bn sin n) then π a f ( ) = + ( ) π a n + b n. efoe deiving the fist majo esult of this pape we need a lemma. In the following we find the infinite seies foms of the definite integals () () and (3)..6. Theoem Suppose is a eal numbe with < π /. Then the definite integals (9) ().5. Lemma Suppose sinh p q ae eal numbes with p + cos q. Then Poof ecause () (7) (8)
6 Ameican Jounal of Numeical Analysis (y Fomula.3.5) = Re n n ( ) (n)! (y DeMoive s fomula) = ( ) (n)! n n n n n (y Eule s fomula) () y Paseval s theoem we obtain n e i(n ) cos(n ). Poof (5) And Similaly because In the following we detemine the infinite seies foms of the definite integals (4) (5) and (6). (y Fomula.3.5) = n n n n ( ) (n)! Also using Paseval s theoem we have sin(n ) (3).8. Theoem Suppose is a eal numbe with definite integals < < π. Then the (6) On the othe hand fom the summation of Eq. (9) and () and using Eq. (8) we obtain (7) efoe deiving the second majo esult of this study we also need a lemma..7. Lemma Suppose sinh p q ae eal numbes with p + sin q. Then Poof ecause (8) (4)
Ameican Jounal of Numeical Analysis 63 (y Fomula.3.6) Using Paseval s theoem we have Similaly because (9) () Net we use Maple to veify the coectness of Eq. (). >evalf(int((sinh(/3*cos())*cosh(/3*cos()))^/((sinh (/3*cos()))^+(cos(/3*sin()))^)^=..*Pi)8);.349545664765686 >evalf(pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(/3)^(4*n- )..infinity)8);.349545664765686 Similaly if = / in Eq. () we have (y Fomula.3.6) Also by Paseval s theoem we obtain. () () >evalf(int((sin(/sqt()*sin())*cos(/sqt()*sin()))^ /((sinh(/sqt()*cos()))^+(cos(/sqt()*sin()))^)^ =..*Pi)8);.66494395547 >evalf(pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(/sqt())^(4*n- )..infinity)8);.66494395547 Finally let = 3/ 4 in Eq. () then In addition fom the summation of Eq. (6) and (7) and using (5) we have 3. Eamples In the following fo the definite integals in this study we povide some eamples and use Theoems and to detemine thei infinite seies foms. On the othe hand we employ Maple to calculate the appoimations of these definite integals and thei solutions fo veifying ou answes. 3.. Eample Taking = / 3 into Eq. (9) we obtain the definite integal (3) >evalf(int(((sinh(3/4*cos()))^+(sin(3/4*sin()))^)/(( sinh(3/4*cos()))^+(cos(3/4*sin()))^)=..*pi)8); 3.66578489849 >evalf(*pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(3/4)^(4*n- )..infinity)8); 3.66578489848 3.. Eample Let = 3 in Eq. (6) we obtain the definite integal (4) >evalf(int((sinh(3*cos())*cosh(3*cos()))^/((sinh(3* cos()))^+(sin(3*sin()))^)^=..*pi)8);.5679593674
64 Ameican Jounal of Numeical Analysis >evalf(pi*(6/9+sum(^(4*n)*(benoulli(*n))^/((*n )!)^*3^(4*n-)..infinity))8);.5679593674 In addition if taking = 5 into Eq. (7) then (5) >evalf(int((sin(sqt(5)*sin())*cos(sqt(5)*sin()))^/(( sinh(sqt(5)*cos()))^+(sin(sqt(5)*sin()))^)^=..* Pi)8);.53964977478 >evalf(pi*(4/45+sum(^(4*n)*(benoulli(*n))^/((*n )!)^*(sqt(5))^(4*n-)..infinity))8);.53964977478 On the othe hand let = 3/ 6 in Eq. (8) then (6) >evalf(int(((sinh(3/6*cos()))^+(cos(3/6*sin()))^ )/((sinh(3/6*cos()))^+(sin(3/6*sin()))^)=..*pi) 8); 5.985398749445 >evalf(*pi*(36/69+69/34)+*pi*sum(^(4*n)*(be noulli(*n))^/((*n)!)^*(3/6)^(4*n- )..infinity)8); 5.985398749446 4. Conclusion In this pape we use Paseval s theoem to detemine some types of definite integals. In fact the applications of this theoem ae etensive and can be used to easily solve many difficult poblems; we endeavo to conduct futhe studies on elated applications. In addition Maple also plays a vital assistive ole in poblem-solving. 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