Integration by Guessing
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1 Itgrtio y Gussig Th computtios i two stdrd itgrtio tchiqus, Sustitutio d Itgrtio y Prts, c strmlid y th Itgrtio y Gussig pproch. This mthod cosists of thr stps: Guss, Diffrtit to chck th guss, d th Adjust to gt ct fit. It is simpl d powrful mthodology, usful i my situtios. Th djustmts r of two kids:. If th guss is off y fctor, divid y th fctor.. If th drivtiv of th guss hs tr trm, th sutrct th itgrl of th trm from th guss (O us is itgrtio y prts). Itgrtio y Gussig mphsizs tht Sustitutio d Itgrtio y Prts gi with hidd guss. Wh doig sustitutio of vrils, choosig u coms log with guss s to th form of th itgrl. Likwis, wh itgrtig y prts, choosig fuctios u d dv corrspods to guss. Rcogizig th rol of hidd gusss lds to shortcuts tht simplify your work, voidig tdious, distrctig clcultios. To simplify positio, ll fuctios mtiod (icludig drivtivs) r cotiuous. D dots diffrtitio with rspct to th vril d w lwys ssum Df ists. Df is lso writt s f or f ( ). rprsts itgrtio d ftr fidig tidrivtiv, w writ fil solutio with ritrry costt C. m d dot positiv itgrs. Thoms [] is our sic rfrc for ottio, cocpts, d tchiqus.. LUCKY GUESSES. My tims you c mk ispird guss to fid idfiit itgrl. Your solutio is justifid y D F() = f() sys th sm thig s F() = f()d, i.., Thorm A (Fudmtl Thorm of th Clculus). If D F() = f(), th f()d = F() Prolm. Fid d. Tmptd to us th sustitutio u = yildig s th solutio, guss Solutio. d =. Diffrtitig, D u s itgrd, pctig. So, = u u As w ll kow, clculus is ot sy Rrly dos ispird guss yild th corrct solutio ctly. Sctios d illustrt how to djust if th prssio producd y guss is clos to wht w wt.
2 . GUESSES OFF BY A CONSTANT FACTOR. If th prssio producd y guss is multipl of wht w wt, w r rscud y th fct tht D is lir oprtor, i.., Thorm B. Prolm.. If DF() = K f(), th. If DF() = Fid f( ) K 5 d. F ( ) f()d C K, th f()d = K F() Tmptd to usd sustitutio, u = 5, yildig guss ( 5 ) Solutio. u s itgrd, pctig. Sic D( 5 ) = 5 ( 5 ). (K=5 i thorm B.. 5 d = Prolm. Fid d. Guss Solutio.. Sic D = d = ( 5 ) 5. (K= i thorm B.), u, w Th t prolm illustrts gussig voidig multipl fussy itrtios of sustitutio. Prolm 4. Fid cos( 5 )d. Guss si( 5 ). Sic Dsi( 5 ) = 0 cos( 5 ). Solutio 4. cos( 5 )d = si( 5 ) 0. GUESSES OFF BY A SIMPLER FUNCTION. If th producd prssio hs tr trm, us Thorm C (D is lir oprtor). If DF() = f() R ( ), th f()d = F() - R()d Rcll tht pplyig Itgrtio y Prts rquirs fidig pproprit u d dv whr
3 Formul D (Itgrtio y prts) (cf. Thoms [], pp ). u dv uv v du..a. INTEGRALS WITH INVERSE FUNCTIONS. Rul.A. To fid I( )d, guss I( ) (corrspodig to u = I(), dv = d). Sic D I( ) I( ) I '( ), ( )d I I( ) - I '( ) d Prolm 5. Fid l d. l l Guss. Diffrtitig, D = l. Solutio 5. l l d = - l d = - 4 Prolm 6. Fid rct( )d Guss rct( ). Diffrtitig, D rct( ) = rct( ) +. l( ) Solutio 6. rct( )d = rct( ) - d = rct( ) -.B. OTHER INTEGRALS WITH MIXED FUNCTIONS. Rul.B. To fid d E( )d, guss E( )d, corrspodig to u =, dv = E() Rul.B works wll wh = d E() is potil typ fuctio lik, si, cos, sih, d cosh. If >, w gt rductio formul d th procss must itrtd. Tulr Itgrtio sms to work ttr (cf. Thoms [], pp 55). Prolm 8. Fid cos( )d. Not cos( )d si( ). Guss si( ). Sic D si( ) = cos( ) + si( ), Solutio 8. cos( )d = si( ) - si( )d = si( ) - ( cos( ))
4 Prolm 9. Fid d. Guss. Diffrtitig, D = +. So, Solutio 9. d = - d = - 4. INTEGRALS EQUAL TO AN INVERSE TRIG FUNCTION Gussig lso provids simpl swrs to itgrls qul to modifid Ivrs Trig Fuctio, limitig fussig with costts. Thorm 4.: Drivd y gussig (Also ot th ic pttr of swrs). rcsi( ) A. d. Guss rcsi( ) rct( ) B. d. Guss rct( ) C. d rcsc( ). Guss rcsc( ). Empl 4.. Fid d. Guss rct( ). Drct( ) Dividig our guss y yilds th rsult., Th gussig tchiqu c usd to prov thorms. This is illustrtd i ppdi which cotis w proof of Ps [] dlig with products of Trig (d Epotil) fuctios. 5. SUPLEMENTARY PROBLEMS. W provid som tr prolms. Hopfully, thy r istructiv d fu to do yourslf. Prolm 4.. Fid 5 d. 5 7
5 Prolm 4. Fid d 9 4 Hit 4.: For d, guss rct( ). Prolm 4.. Fid l d Prolm 4.4. Fid rct( )d Prolm 4.5. Fid Prolm 4.6. Fid si( )d. 5 l d. Prolm 4.7. Fid ( l + )d. Hit 4.7: Do ot pic. Eplor th proprtis of,.g., D l +. REFERENCES. R.L. Fiy, F.R. Giordo, d M.D. Wir, Thoms Clculus,0 th d., Addiso Wsly Logm, Bosto, D.K. Ps, A usful itgrl formul, Amr. Mth. Mothly 66 (959) 908. APPENDIX. PEASE S THEOREM RESURRECTED. Itgrls of products of trig d potil fuctios. Dfiitio 5.: A fuctio f() is qusi-potil if f '' hf for som h. h is th itrtio strgth of f writt [ f] h. Thorm 5.: Th followig fuctios H() r qusi-potil: A. [ ], [si ] [ cos ], [sih ] [cosh ] d [ ] 0 B. [H( )] [H()]. C. I prticulr, [ ], [si ] [ cos ], [sih ] [cosh ] Usig Itgrtio y prts, itrtd twic, Ps [] hs show Thorm 5. (Ps): If f d g r qusi-potil with f ' g f g ' f ' g f g ' th f g = C = C. h k [ f ] [ g] f '' hf d g '' kg (h k),
6 Proof 5.(By Gussig): Thr r two turl gusss: f g d fg. Guss f ' g. Df ' g f g f g. So, Df ' g hfg f g (sic f '' hf ). Guss f g'. Df g ' fg '' f g. So, Df g ' kfg f g (sic g'' kg). Which guss do w us? W us oth, tkig systm of qutios viw. Df ' g Df g ( h k)f g (sutrctig guss computtios). So f ' g f g ( h k) f g (itgrtig). Prolm 5.4. Fid si()d. Tkig ( ) f d g( ) si( ): f '( ), g '( ) cos( ), h, k si() - cos() Solutio 5.4. si()d =. ( ) Prolm 5.5. Fid si() cos( )d ( ). Tkig f ( ) si( ) d g( ) cos( ): f '( ) cos( ), g '( ) si( ), h, k Solutio 5.5. cos() cos() si()si () si() cos()d = ( ) Not si() cos( )d, si() cosh( )d, sih() cosh( )d r foud similrly. Corollry 5.6: If f is qusi-potil, Prolm 5.7. Fid Solutio 5.7. [si( )] si() d. f = f ' [ f ]. So, si() d f + C cos()-si() =
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