LASS XII Q Evlut : Q sc Evlut c Q Evlut: ( ) Q Evlut: Q5 α Evlut: α Q Evlut: Q7 Evlut: { t (t sc )} / Q8 Evlut : ( )( ) Q9 Evlut: Q0 Evlut: Q Evlut : ( ) ( ) Q Evlut : / ( ) Q Evlut: / ( ) Q Evlut : ) ( Q5 Evlut: Q Q7 Evlut ( α ) ( α) Evlut: ( ) log 5 ( ) Evlut: 5 Q8 Evlut: Q9 Evlut: ( ) Q0 Evlut ( ) Q Q Evlut: ( ) Trgt Mthmtics by- Agyt Gupt t Q Evlut: θ θ Q θ log θ θ θ sol: ( Evlut: ) Q5 Evlut : t Q Evlut: Q7 t t Evlut: Q8 Q9 Q0 LASS XII Evlut: Evlut: Evlut: 7 0 Q Evlut: ( log ) Q Evlut: ( ) Q ( ) 5 Evlut: 5 ( ) Evlut: ( ) Q Evlut: Q5 Q Q7 Q8 Q9 Q0 Q φ φ Evlut : φ φ Evlut: Evlut : Evlut : φ 9 ( ) Evlut: Evlut: Q Evlut : Si Q Evlut : ( ) Q Q5 Q Q7 Q8 Q9 7 log Evlut: Evlut: ( ) 8 Evlut : 7 Evlut : log( Evlut : [ ] Evlut : Trgt Mthmtics by- Agyt Gupt ) log ( ) \ Q50 Evlut : Q5 Q5 Evlut : [ ] Q5 Q5 Q55 Evlut ( ) / cot (cot c) Evlut: 5 Evlut: t log Writ vlu of ( ) Q5 Evlut :
Q57 Q58 Q59 Q0 Q Q Q Q LASS XII Evlut: (t cot ) Evlut: Evlut: / ( ) Evlut: 5 Evlut: Evlut: 5 7 Evlut : 0 Evlut: ( b ) Q5 Evlut: b Q Q7 Q8 Q9 Q70 ( c Evlut : ( )( Trgt Mthmtics by- Agyt Gupt )( Evlut : Evlut : ( ) Evlut ( ) ( ) ) ) Evlut ( )( )( ) log Q7 Evlut: ( ) Q7 Q7 Q7 Q75 Evlut: sc ( )( Evlut: Evlut: ) log(log ) (log ) Evlut : Q7 log Evlut : ( log ) Q77 Evlut : ( ) Q78 Q79 Q80 Q8 Q8 Evlut: Evlut: os os( α) Evlut: 9 Evlut: Evlut: ( ) LASS XII Q8 Evlut: b Q8 ( ) Evlut: ( ) Q85 Evlut: c log t Q8 Evlut : [(log ) log ] Q87 Evlut: Q88 Evlut: ( )( ) Q89 Evlut: Q90 Evlut : sc Q9 Evlut: Q9 Evlut: Q9 Evlut: ( ) Q9 Evlut : ( ) ( ) Q95 Evlut : Q9 Q97 Q98 Q99 Q00 9 Evlut: Evlut : 9 Evlut : Evlut : Evlut: { } (log ) 7log t Q0 Evlut: Q0 Evlut: Q0 Evlut: log Q0 Evlut: Q05 5 log log Evlut : log log Q0 Evlut: Q07 Evlut : t t t Q08 Q09 Q0 Evlut: ( Evlut : t / ) Evlut 9 Trgt Mthmtics by- Agyt Gupt
Q Q Q Q Q5 Q Q7 Q8 Q9 Q0 Q Q Q Q Q5 Q Q7 Q8 Q9 Q0 Q Q Q Q Q5 LASS XII Evlut : Evlut : 9 Evlut : log log(log ) 5 Evlut : Evlut : Evlut : Evlut : t p Evlut: sc t Evlut : Evlut : 5 Evlut : Evlut: t Evlut : Evlut: Evlut: Evlut: Evlut: / Evlut : Evlut Evlut: ( ) ( ) 5 ( )( b) ( ) / ( ) Evlut : Evlut: / ( ) ( ) t Evlut: t 8cot Evlut: cot Evlut: / LASS XII Q Evlut : t Q7 Evlut : 5 Q8 Evlut : Q9 Q0 Q Q Q Q Q5 Q Q7 Q8 Q9 Evlut : Evlut : Evlut: t ( α) ( α ) Evlut : ( α ) Evlut : Evlut : ( ) Evlut : Evlut : 5 Evlut: 5 Evlut: Evlut ( ) Q50 Evlut: log (log ) Q5 Evlut 7 0 Q5 Evlut Q5 Q5 Q55 Q5 Q57 Q58 Q59 Q0 Q ( ) Evlut: ( ) Evlut: Evlut: 8 Evlut: Evlut: Evlut : ( ) t Evlut : Evlut: Evlut ( )( )( ) Trgt Mthmtics by- Agyt Gupt Trgt Mthmtics by- Agyt Gupt
LASS XII Q Evlut ( )( ) Q Evlut ( )( ) Q Evlut ( )( ) Q5 8 Evlut ( )( ) Q Evlut ( ) ( ) Q7 Evlut ( ) Q8 Evlut ( ) ( ) Q9 Evlut ( ) Q70 Evlut ( ) Q7 θ Evlut θ ( θ )( θ ) Q7 Evlut ( )( ) Q7 Evlut ( ) Q7 Evlut ( ) ( ) Q75 Evlut Q7 Q77 Q78 Q79 Evlut sc c Evlut ( ) Q80 Evlut { } Q8 Q8 Evlut Evlut t θ cotθ θ Evlut 5 Evlut ( )( )( ) Q8 Evlut c Q8 Evlut 5 ( ) Q85 t sc Evlut t Q8 Evlut Q87 Evlut 5 Trgt Mthmtics by- Agyt Gupt Q88 Q89 Q90 Q9 Q9 Q9 LASS XII Evlut Evlut sc Evlut 5 Evlut Evlut Evlut Q9 Evlut Q95 ( ) Evlut ( ) Q9 / Evlut Q97 Evlut t Q98 cot Evlut: log Q99 Evlut Q00 Evlut t Q0 t Evlut: ( ) Q0 log Evlut: ( ) Q0 Evlut (/ ) Q0 Q05 Q0 Q07 Q08 Q09 Q0 Q Q Q Evlut ( ) Evlut b Evlut: Evlut : / Trgt Mthmtics by- Agyt Gupt Evlut : Evlut : ( )( b) Evlut: cot θ cot θ Evlut : cot θ
Q Q5 Q Q7 Q8 Q9 Q0 LASS XII **********//************* FUNDAMENTAL INTEGRATION FORMULAS = (i), (iii) = Trgt Mthmtics by- Agyt Gupt (log ) = = log ( ) = = log (iv) =, = log (v) (vi) (vii) (viii) ( ) = = ( ) = = (t ) = sc sc = t ( cot ) = c c = cot (i) (sc ) = sc t sc t = sc () ( c) = ccot ccot = c (i) (log ) = cot cot = log (iii) (iv) t ( log ) = t = log (log(sc t )) = sc sc = logsc t (log( c cot )) = c LASS XII c = log c cot (v) = = (vi) = = (vii) t = = t (viii) (i) cot = = cot sc = = sc () c = Trgt Mthmtics by- Agyt Gupt = c Som Importt Itgrtios ( b) = log b b (i) ( b) =, (iii) b = b b b c log (iv) b c =, > 0 ( b) = ( b) sc ( b) = t( b) c ( b) = cot( b) sc( b) t( b) = sc( b) c ( b) cot( b) = c( b) t( b) = log ( b) cot( b) = log ( b) sc( b) = log sc( b) t( b) (v) ( b) = ( b) (vi) (vii) (viii) (i) () (i) (iii) (iv) c ( b) = log c( b) cot( b) Som Spcil Itgrls (i) = t = log
LASS XII (iii) = log (iv) = (v) = log Trgt Mthmtics by- Agyt Gupt (vi) = log Som Importt Itgrls (i) = = log = log Importt Not: To vlut itgrls of th form m, m, m m, w us th followig trigoomtricl ititis Ititis: A B = ( A B) ( A B) : A B = ( A B) ( A B) A B = ( A B) ( A B) : A B = ( A B) ( A B) m Itgrls Of Th Form, Whr m, r Positiv Itgrs I th itgrls of th form m th followig substitutios r usful (i) If m is o i, powr of is o, put = t If is o i, powr of is o, put = t (iii) If both m r v, th us D Moivr s thorm Som Importt Substitutios Followig r som substitutios usful i vlutig itgrls Eprssio Substitutio = tθ or cotθ = θ or θ = scθ or cθ or = θ Itgrls Of Th Typ b c To vlut this of itgrls w prss b c s th sum or iffrc of two squrs by ug th followig stps STEP I Mk th cofficit of uity by tkig it commo STEP II A subtrct th squr of hlf of th cofficit of Itgrls Of Th Typ b c To vlut this typ of itgrls, w prss b c s th sum or iffrc of two squrs by ug th followig stps STEP I Mk th cofficit of uity by tkig it commo STEP II A subtrct squr of hlf of th cofficit of p q Itgrls Of Th Form b c To vlut this typ of itgrls w prss th LASS XII umrtor s follows: p q = λ (Diff of omitor) µ = λ ( b) µ P( ) Itgrls Of Th Form b c Whr P(X) is Polyomil of Dgr Grtr Th or Equl To To vlut this typ of itgrls w ivi th umrtor by th omitor prss th itgr s R( ) Q( ) b c whr R() is lir fuctio of P( ) R( ) = Q( ) b c b c p q Itgrls Of Th Form b c To vlut this typ of itgrls w prss th umrtor s follows p q = λ (Diff of omitor) µ = λ( b) µ whr λ µ r tts to b trmi by qutig th cofficits of similr trms o both sis So w hv Itgrls Of Th Form, b,,, b b ( b ) b c To vlut this typ of itgrls w procc s follows STEP I Divi umrtor omitor both by STEP II Rplc sc, if y, i omitor by t STEP III Put t = t so tht sc = t Itgrls Of Th Form, b,, b b b c To vlut this typ of itgrls w proc s follows t / STEP Put =, t / STEP Rplc STEP Put t = t t i th umrtor by t = t so tht sc = t b c / / Trgt Mthmtics by- Agyt Gupt sc Itgrls Of Th Form To vlut this typ of itgrls w prss th umrtor s follows Numrtor = λ(diff of omitor) µ (omitor) λ ( c ) i ( b ) = ( c ) µ whr λ µ r tts to b trmi by comprig th cofficts of o both sis b c λ( c ) µ ( c ) = c c = µ λ = µ λlogc K c
LASS XII b c Itgrls Of Th Form p q r To vlut this typ of itgrls, w prss th umrtor s follows Numrtor = λ (omitor) µ (Diff of omitor) V i, ( b c) = λ ( p q r) µ ( p q ) v Whr λ,µ, v r tts to b trmi by comprig th cofficits of, tt trm o both sis b c p q r Diff of omitor = λ µ omitor v p q r = λ µ logomitor v p q r INTEGRATION BY PARTS Thorm: If u v r two fuctios of, th u uv = u( v) v i th itgrl of th prouct of two fuctios = (First fuctio) (Itgrl of sco fuctio) itgrl of {(Diff of first fuctio) (itgrl of sco fuctio)} Not Propr choic of first sco fuctio Itgrtio with th hlp of th bov rul is cll th itgrtio by prts I th bov rul thr r two trms o RHS i both th trms th itgrl of th sco fuctio is ivolv Thrfor i th prouct of two fuctios if o of th two fuctios is ot irctly itgrbl (g, log, -, t - tc) w tk it s th first fuctio th rmiig fuctio is tk s th sco fuctio If thr is o othr fuctio, th uity is tk s th sco fuctio If i th itgrl both th fuctios r sily itgrbl, th th first fuctio is chos i such wy tht th rivtiv of th fuctio is simpl fuctio th fuctio thus obti ur th itgrl sig is sily itgrbl th th origil fuctio Not W c lso choos th first fuctio s th fuctio which coms first i th wor ILATE, whr I Sts for th ivrs trigoomtric fuctio (,,t tc) L Sts for th logrithmic fuctios A Sts for th lgbric fuctios T Sts for th trigoomtric fuctios E Sts for th potil fuctios Itgrls of Th Form { f ( ) f '( ) } = = f ( ) f '( ) II I = f ( ) f '( ) f '( ) = f ( ) Itgrls Of Th Form ( p q) b c To vlut itgrls of th typ ( p q) b c, w prss th lir fctor p q s follows p q = λ ( b c) µ Itgrls of Th Form, λ LASS XII, λ λ Whr λ is ostt To vlut this typ of itgrls, ivi th umrtor omitor by put = t or = t, which vr or iffrtitio givs th umrtor of th rsultig itgr Itgrtio Of Som Spcil Irrtiol Algbric Fuctios I this rticl w shll iscuss four itgrls of th φ() form, whr P Q r polyomil fuctios of φ( ) Itgrls of th form, whr p q both r lir fuctios of To vlut this typ of itgrls w put Q = t i to vlut itgrls of th form put c = t ( b) c φ() Itgrls Of Th Form, Whr P Is A Qurtic Eprssio A Q Is A Lir Eprssio To vlut this typ of itgrls w put Q = t i, to vlut itgrls of th form, put p q = t ( b c) p q φ( ) Itgrls Of Th Form, Whr P Is A Lir Eprssio A Q Is A Qurtic Eprssio To vlut this typ of itgrls w put p = / t i to vlut itgrls of th form ( b) p q r, put Itgrls Of Th Form b = t φ( ), Whr P A Q Both Ar Pur Qurtic Eprssio I i P = b Q = c To vlut this typ of itgrls w put t = th c t = u i to vlut itgrls of th form ( b) c t tt = to obti bt ) c ( t, w put th c t = u Trgt Mthmtics by- Agyt Gupt Trgt Mthmtics by- Agyt Gupt