CLASS XII CBSE MATHEMATICS INTEGRALS

Transcription

1 Using Partial Fractions LSS XII SE MTHEMTIS INTEGRLS () cos ( sin)(sin ) () ns: log sin sin () () (SE 8) tan (sin ) c Let sin t cos ( t)( t ) t ( )( ) cosθ (sin θ)(5 cos θ) t,,, t (SE 8 OMP) dθ (SE 7) (5) ns: tan ( sinθ ) 5 tan (sinθ) (6) (SE 7) ( )( ) ns: log log Let t, e, (t).(t) t t (7) (e )(e (SE 7) ) ns: 8 log e 6 log e 8 tan ( e ) c (8) (9) ( )()( ) ( )()( ) ( )( ) ( )( ) () ()( ) () ()( ) () () () Evaluate : Hint : () Evaluate D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY (t) (ns: 5 log 5 log 5 tan ),,, (ns: () () () ()( ) ( ) (). ( ) () ( ), log 5 () ), 5,

2 () Evaluate : ( )( ) Let t,, (5) Evaluate : ( )( ) (t)(t) t t Let t,,, I (6) Evaluate: 8 (7) Evaluate (8) 8 (8) 8 cos ( sin )( sin ) Let sin t, cos, (8) Evaluate (e )(e ) Let e t e, e ( t)( t) I (t)(t) (ns: log sin (t)(t) ) sin,, t t (9) (sinφ )cosφ dφ (ns: log ( sin ) ) 5 cos φ sin φ sin Let sin, then I ( ), ( ) ( ),, Improper Rational Functions () ( )( ) Divide Numerator by denominator, we get ( )( ) 56 () (ns: 5log log ) 5 5 ( 5 ) ( ) ( )( ) Divide Numerator by denominator, we get ( )( ) ( ) ( ) () Evaluate the following: () ( ) ( )( ) ( )( ) (ns: log log ) ( ) ( ) 5 ( ) () 5 5 D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY ()() (ns: log log( ) tan ) Divide Numerator by denominator, we get ( )( ) ( ) ( )( ),,,

3 Linear/Quadratic or Linear/ Quadratic or Linear. Quadratic () ns: 5 8 () () () log () (SE 7) () ns: sin ( ) (SE 7) (ns: [( 6 ) ( 7 (5) (6) 5 ( ) 5 5. (7). 5 (8) (9) () ( )( ) 67 ( 5)( ) 5 () 5 () ( ) 7 log 5, where 7 log ) 6 ) ] e () 5 e e Integrals using the formula e {f() f ()} e. f() c () Evaluate ( )e ( ) () Evaluate: ( )e ( ) sin () Evaluate ( cos e () Evaluate: e ( sin cos (5) Evaluate: e ( sin cos ) Integration by Parts () e sin( ) ns: e { sin( ) cos ( )} (SE 7) () Evaluate: sin () (tan ). D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY

4 () sin (5) sin (6) cos (7) (sin ) (8) tan (9) [log(log ) (log ) ] () Evaluate : (tan ). Integral as a limit of Sum () Evaluate: ( e ) () Evaluate ( ) as the limit of a sum () Evaluate : ( 5) as limit of sums () Evaluate: ( ) (5) Evaluate : 5 as the limit sum (SE 8) as limit of sum as a limit of a sum (6) Evaluate ( ). as limit of sums. (7) Evaluate: ( ), using limit of sums Properties of Integrals sin cos () Evaluate sin cos tan () Evaluate : ns: ( ) sec tan sin cos / () Evaluate: { } ns: () Evaluate: (5) Evaluate: sin (SE 8 OMP) (SE 8 OMP) (SE 7) (SE 7) ns: (SE 7) (SE 7) / ns: sin sin ns: ( ) ( ) ( ) (6) Evaluate: (7) Evaluate log( tan) / (8) Evaluate: logsin. (9) Evaluate: log ( cos ) () Evaluate: ( logsin log sin ) () Evaluate : a cos b sin (SE 7) ( ns: ab ) D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY

5 () Evaluate: () Evaluate : () Evaluate: sin cos sin. cos sin a (5) Evaluate: sin a a (6) Evaluate: a a a (7) Evaluate: cot ( ) (8) Evaluate : (9) Evaluate / /6 tan 5. () Evaluate: f(), where f() 5 () sin( ) Other Integrals ns: () Evaluate: sinsos 69sin / () Show that: ( tan cot) () ( tan cot) () sin cos tan (sin ) (5) sin. sin. sin (6) sin cos sin 6 (7) Evaluate: Sin( a) Sin(a) sin (8) Evaluate: sin (a) (9) Evaluate tan sin sin () Evaluate tan () Evaluate: 5 cos () Evaluate: [log( ) log ] () Evaluate () Evaluate : tan (5) (6) tan cos cos α cos cos α (SE 8) D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY 5

6 (7) cos( a)cos( b) Mark / Marks Questions () Find: 5 sin cos (SE 8 OMP) () Evaluate: cos sin (SE 8) cos () Find : sin cos (SE 7) sincs ns: log sin c or log sec log sin c () Evaluate: ns: 8 log (5) Evaluate cos 5 ns: Zero (6) Evaluate: (cos sin ) (7) Write the value of : (8) e (tan ) (9) Find the value of sin () Evaluate () Write the value of 6 () Evaluate: 7 (SE 7) (SE 7) ( ns : Zero ) () Write the value of e log. ( ns: 8 ) 8 () Evaluate: e log e log e log elog ( ns: ) (5) Evaluate e ( log ). (6) Evaluate: cos sin. (7) Evaluate : sin (8) Evaluate : (9) Evaluate : () Evaluate: () Evaluate: cos6 sin6 D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY 6

7 () Evaluate: / () Evaluate: cos () Evaluate: sin.. (5) Evaluate: sec. (6) Evaluate: sec tan. (7) If ( k),find the value of k (8) Evaluate: ecos e cos e cos (9) Evaluate sec (7 ). () Evaluate: () Evaluate: (log ) log.. () Evaluate: ( ). () Evaluate : (log ) () Evaluate :. D SREENIVSULU, M.Sc,M.Phil,.Ed, PGT(MTHEMTIS), KENDRIY VIDYLY 7