SIMPLIFIED METHOD OF EVALUATING INTEGRALS OF POWERS OF COSINE USING REDUCTION FORMULA AS MATHEMATICAL ALGORITHM Lito E. Suello Malaya Colleges Lagua, Pulo-Diezmo Road, Cabuyao City, Lagua, Philippies Correspodig Author: lesuello@mcl.edu.ph Abstract A simpler ad shorter method of evaluatig the itegrals of powers of ie is preseted i this paper. Geeralized formulas i evaluatig itegrals of odd ad eve powers of ie were derived by repeatedly applyig the reductio formula for ie to the itegral of the th power of ie. From the behavior of the coefficiets ad expoets of the terms of the derived formulas, algorithms were developed. The ew method was compared with the traditioal method ad results showed that the ew method was simpler ad shorter. The derived formulas ad developed algorithms will be very useful i the study of higher mathematics courses ad i may egieerig applicatios. Keywords: Itegratio, Mathematical Algorithm, Powers of Cosie, Reductio Formula, Trigoometric Idetities Itroductio Oe of the essetial topics i the study of Itegral Calculus is evaluatig itegrals of powers of trigoometric fuctios. The traditioal methods usually use trigoometric idetities to trasform powers of trigoometric fuctios ito a form where direct itegratio formulas ca be applied. The idetity used depeds o whether the power is odd or eve. For odd powers of ie, the idetity x si x is applied. The itegrad is trasformed by factorig out oe ie ad the remaiig eve powered ie is coverted ito sie usig the idetity. The itegral is the evaluated usig a power formula with the factored ie used as the differetial of sie. For eve powers of ie, the double agle idetity x ( x) is used to reduce the power of ie ito a expressio where appropriate itegratio formulas ca already be applied (Dampil, 04; Dampil, 05; Hass, Weir, & Thomas, 04). Aother method used to evaluate powers of ie is by usig a reductio formula. A reductio formula trasforms the itegral ito a itegral of the same or similar expressio with a lower iteger expoet (Riley, Hobso, & Bece, 00). It is repeatedly applied util the power of the last term is reduced to two or oe ad the fial itegral ca be evaluated. Usig itegratio by parts, the reductio formula for ie is (Stewart, 04). ax si ax a ( ) Page6 The methods discussed above are ormally tedious ad time umig depedig o the give power of ie. As show i the studies of Dampil (Stewart, 0; Suello, 05), derivig geeralized formulas ca simplify solutios. The study of Varberg, Purcell ad Rigdo (04) also revealed that the reductio formula for sie ca be geeralized ad a simpler algorithm ca be developed to evaluate itegrals of powers of sie. The objective of this paper is to exted the same cocept to the itegrals of powers of ie. Geeralized formulas were derived by the repeated applicatio of the reductio formula to the itegral of Asia Pacific Istitute of Advaced Research (APIAR)
the th power of ie. The behavior of the coefficiets ad expoets of the terms i the derived formulas were used as the basis for developig a simpler algorithm. Derivatio of Formulas Give:, where is ay iteger Usig the reductio formula, ) ax si ax Applyig the reductio formula to the last term ) axsi ax ) 3 ax si ax 3 4 Applyig the reductio formula agai, ax si ax si ) )( ) 3 ax ax ( )( 3) ( ) 4) 3 axsi ax 4 5 6 Simplifyig, ) ( )( 3) )( )( 4) ax si ax )( ) 3 ax si ax ( )( 3) axsi ax ( )( 4) 5 6 The same tred cotiues util the last term becomes if is odd, or if is eve ). Odd Powers 3 ax si ax axsi ax )( ) Page7 )( 3) )( )( 4) ( 5 ( )( 3)( 5)...() axsi ax... ( )( 4)( 6)...(3) Asia Pacific Istitute of Advaced Research (APIAR)
Itegratig the last term, ) )( 3) )( )( 4) ( 5 ax si ax )( ) 3 axsi ax )( ) 3 ax si ax ( )( 3)( 5)...() axsi ax... si ax C )( )( 4)( 6)...(3) Factorig out the commo factor gives the formula, si ax a ax ( ) ( )( 3)( 5)...() C ( )( 4)( 6)...(3) It ca also be writte as si ax C ax C j a 0 j 3 j ( )( 3) ax ( )( 4) ax C 5 ax... where: C 0 ad C j C j j j. Eve Powers ) ( )( 3) )( )( 4) ax si ax )( ) 3 axsi ax 5 ( )( 3)( 5)...(3) axsi ax... ( )( 4)( 6)...(4) Applyig the reductio formula to the last term, Page8 ) axsi ax )( ) ( )( 3) axsi ax )( )( 4) 3 5 ax si ax Asia Pacific Istitute of Advaced Research (APIAR)
Simplifyig, ( )( 3)( 5)...(3)... axsi ax ( )( 4)( 6)...(4) ) ) 0 axsi ax )( ) ( )( 3) axsi ax )( )( 4) 3 5 ax si ax ( )( 3)...(3) ( )( 3)...(3)... axsi ax x C )( )( 4)...() )( )( 4)...() Factorig out the commo factor gives, si ax a ax ( ) 3 ( )( 3) ax ( )( 4) ( )( 3)( 5)...(3) ( )( 3)( 5)...(3) ax x C ( )( 4)( 6)...() ( )( 4)( 6)...() 5 ax... The formula may also be writte as, si ax C ax C j a 0 j j ax C x C where: C 0 ad C j C j j j 3. Developmet of the Algorithm for the New Method A simpler ad easier procedure ca be developed from the observed treds of the coefficiets ad expoets of the derived formulas. These are summarized as follows: 3. Odd Powers Page9 Write si ax a 5 xdx. This will be followed by a series of ie terms. For example, si x Asia Pacific Istitute of Advaced Research (APIAR)
The first term of the series has a coefficiet of ad the expoet of ie is -. This coefficiet ad expoet will be used i determiig the coefficiet ad expoet of the ext term. si x 4 x 5 For the ext term, the coefficiet has a umerator equal to the product of the expoet ad the umerator of the precedig term. The deomiator is the product of the deomiator ad expoet mius oe of the precedig term. The expoet of ie is the expoet of the precedig term mius two. si x 4 ()(4) x x 5 (5)(3) Follow the same procedure util the expoet of ie becomes zero which termiates the series. x 4 4 (4)() 0 x x x 5 5 (5)() Add a tat of itegratio. 5 si x 4 4 5 xdx x x C 5 5 5 3. Eve Powers si ax Write a. This will be followed by a series of ie terms. For example, 6 3xdx si 3x 3 The first term of the series has a coefficiet of ad the expoet of ie is -. This coefficiet ad expoet will be used i determiig the coefficiet ad expoet of the ext term. si 3x 5 3x 3 6 For the ext term, the coefficiet has a umerator equal to the product of the expoet ad the umerator of the precedig term. The deomiator is the product of the deomiator ad expoet mius oe of the precedig term. The expoet of sie is the expoet of the precedig term mius two. si 3x 5 ()(5) 3 3x 3x 3 6 (6)(4) Page0 Follow the same procedure util the expoet of ie becomes oe which termiates the series. Asia Pacific Istitute of Advaced Research (APIAR)
si 3x 5 5 3 (5)(3) 3x 3x 3x 3 6 4 (4)() The ext term is the product of x ad the coefficiet of the last term i the ie series. si 3x 5 5 3 5 5 3x 3x 3x x 3 6 4 48 48 Add a tat of itegratio. 6 si 3x 5 5 3 5 5 3xdx 3x 3x 3x x C 3 6 4 48 48 4. Compariso betwee the Old ad the New Method Evaluate 7 4xdx. Usig the Old Method 7 6 6 5 5xdx 5xsi 5x 5xdx 5(7) 7 6 6 4 4 5xsi 5x 5xsi 5x 3 5x 35 7 5(5) 5 6 6 4 4 5xsi 5x 5xsi 5x 5xsi 5x 5xdx 35 75 35 5(3) 3 6 6 4 4 48 5xsi 5x 5xsi 5x 5xsi 5x si 5x C 35 75 55 55 si 5x 6 6 4 8 6 5x 5x si 5x C 5 7 35 35 35 Usig the New Method 7 si 5x 6 ()(6) 4 (6)(4) (4)() 0 5xdx 5x 5x 5x 5x C 5 7 (7)(5) (35)(3) (05)() 7 si 5x 6 6 4 8 6 5xdx 5x 5x si 5x C 5 7 35 35 35 Page Evaluate 4 x Asia Pacific Istitute of Advaced Research (APIAR)
Usig the Old Method 4 3 3 xdx xsi x xdx (4) 4 xdx 8 4 3 3 xsi x 4 () xsi x 4 3 3 3 xdx xsi x xsi x x C 8 6 8 4 si x 3 3 3 xdx x x x C 4 8 8 dx Usig the New Method 4 si x 3 ()(3) 3 xdx x x x C 4 (4)() 8 4 si x 3 3 3 xdx x x x C 4 8 8 5. Coclusio The ew algorithms developed revealed that the itegrals of powers of ie ca be evaluated easily sice the tedious repetitios of applyig the reductio formula, or expasios of idetities usig the traditioal methods, are elimiated. Itegrals ca be evaluated directly sice the procedure simply ivolves coefficiets ad expoets. The derived formulas ad algorithms will be very useful i higher mathematics courses like Differetial Equatios ad Advaced Egieerig Mathematics ad eve i the fields of Physics ad Mechaics. It ca also be used i may egieerig applicatios specifically i electricity ad magetism, waves, heat ad mass trasfer ad reactio kietics. It is also recommeded that the procedure also be applied to the itegrals of powers of other trigoometric fuctios. Page Asia Pacific Istitute of Advaced Research (APIAR)
Refereces Dampil, F. 04. Developmet of trigoometric formula for si 4 ѳ ad 4 ѳ usig half-agle idetities as mathematical algorithm, Iteratioal Joural of Applied Physics ad Mathematics, 4 (3), 55-58. Dampil, F. 05. A simplified formula i reducig the powers of taget ad cotaget usig squares of taget theorem as the iitial approach, upublished article, Malaya Colleges Lagua. Hass, J., Weir, M. ad Thomas, G. 04. Uiversity Calculus, Pearso, 439-440. Riley, K.F., Hobso, M.P. ad Bece, S.J. 00. Mathematical Methods for Physics ad Egieerig, Cambridge Uiversity Press. Stewart, J. 04. Calculus: Early Trascedetals, 7ed, Cegage, 467-468. Stewart, J. 0. Calculus: Cocepts ad Cotexts, 4ed, Cegage, 389-390. Suello, L. 05. Simplified method of evaluatig itegrals of powers of sie usig reductio formula as mathematical algorithm, Iteratioal Joural of Applied Physics ad Mathematics, 5 (3), 9-98. Varberg, D., Purcell, E. ad Rigdo, S. 04. Calculus Early Trascedetals, st ed., Pearso, 385-386. Lito E. Suello is a member of the Mathematical Society of the Philippies ad the Philippie Istitute of Chemical Egieers. Bor i Camaries Sur, Philippies o February, 966, he is a licesed chemical egieer who fiished his chemical egieerig degree from Cetral Philippie Uiversity i Iloilo City, Philippies i 987. He also completed master i busiess admiistratio from Sa Pedro College of Busiess Admiistratio, Lagua, Philippies i 006. At preset, he is coected with Malaya Colleges Lagua, Philippies where he teaches mathematics, mechaics, ad chemical egieerig courses. Page3 Asia Pacific Istitute of Advaced Research (APIAR)