CHAPTER:Idfiit Itgrals Multipl Choi Qustios with solutio: 0. d = a )log( ) + b )log( ) + )log( ) + + d ) N o As : Put dt = t: d= dt d= ( t + ) dt = d = = dt ( t+ )( t) t t+ = log( ) log( ) + = log( ) + As :( a) 0. (ta + ot d ) = a)logta+ b)log(si + os ) + )log+ No Solutio : (ta + ) s I = = d ta ta = log(ta ) + As :( a)
0. si(log ) + os(log ) d= a ) si(log ) + b ) os(log ) + ) si(log ) os(log ) + o Solutio : Itgratio. by. parts si(log ).os(log ). d+ os(log ) d si(log ) + As : a) 0. log log a) + log b) + )log + log d = + log Solutio : d = d = log + As :( ) ta 05. d = a ) ta+ b)/ ta(/ ) + )/ + ta(/ ) + o Solutio : put / = = / = / t t d t dt = t ta t + As :( b) ta (s ) I = tdt = t dt
06. ta d = a)log(os ) + b)log(s ) + ) log(s ) + o Solutio : put t = dt = / d dt = d I = ta tdt = log s + As :( b) 07. 8 d + 8 a)log( + ) + b)ta ( + ) + )ta )ta d Solutio : put t = th + + = ta + I
08.. If k = a)log b)/ log )(log ) d k.si ( ), th / Solutio : = t =.logd = dt dt si ( I = ) log = + t log As :( ) 09. os d = (si + os ) a) + si + os b)log(si + os ) + )log(si os ) + )log(si+ os ) + d Solutio : I = = log(si + os ) + As : b) os si d (si + os )
0. d = a) + b) + ) + + Solutio : I = d. = = + As :. ( + ) d = ( + 7) a) 9 b) log(+ 7) + 9 9 ) log(+ 7) + 9 log(+ 7) + a + b a ( ad b) Solutio : d = log( + + d (8 9) I = log( + 7) + 9 9 I = log( + 7) + 9 As :( )
.. d= a) ( + log ) + log b) (log ) + log ) (.log ) + (log ) o Solutio : Itgratio. by. parts. I =.. = (.log ) + log (log ) (log ) As :( ) 5. d = + a) + + ta + b) + log( + ) + ) + ta + ta + Solutio :( Divisio) I = ( + ) d + log( I = + + ) + As :( b)
( + ). ( ) a ). b ) + + )( + ) + ( ) + + d ( + + ) Solutio : ( ). d I = ( + ) + d 5. = 7+ 5os a) ta ta + b) ta ta 6 6 + ) ta (ta / ) + 7 ta (ta /) + a b Solutio : d = ta ta a+ bos a b a+ b 9 5 = = ta ta + 7+ 5os 9 5 I d I = ta ta / + 6 6 As( b)
6. (si os ) os a) + si b) + si ) + os + d= Solutio + d : (si os )(si os ) si I = os d= + As :( b) ( a + b ) 7. d = ab ( a/ b) ( b/ a) a) + + + log ( a/ b) log ( b/ a) ( a/ b) ( b/ a) b) + + log( a/ b) log( b/ a) )( a/ b).log( a/ b) + ( b/ a) log( b/ a) + + ( a/ b).log( a/ b) ( b/ a) log( b/ a) + Solutio : ( a + b + a. b a b d = ( ) a. b + + d b a ( a/ b) ( b/ a) = + + + log ( a/ b) log ( b/ a) As :( a)
os 8. d = + 9ot ot a) ta + ot b) ta + 6 ) log( + 9 ot ) + 9 ot ta + 6 Solutio : put ot t d dt = os = dt / ot + t 6 As :( 9. If = ta + d = + 5+ os th k.ta ( M.ta /) ak ) = bk ) = / M ) = / dm ) = / d Solutio : I = = ta / 5+ ta + / = = t ta /, dt / s / d dt ta / t + As : k = /:( b) = = ta + /.s / 9+ ta / d
0. os d a) ( os.ot ) + log(os+ ot ) + b) (os.ot ) + log(os+ ot ) + ) (os.ot ) log(os + ot ) + ( os.ot ) log(os + ot ) + Solutio : os =.os I d = I = os.ot ot.os d I = os.ot (os ). osd I = os.ot I + log(os + ot ) + os.ot log(os + ot ) I = + + As :( a)
. d = g( ) + a whr g ( ) = a)/os b)/.si )/.si d ) /.os ( / ) Solutio : Put / a = siθ a a a / d = a.osθdθ / osθdθ si I = a (si θ ) = + / a si θ s = i a As :( ) a a / +
. (log ) d th I a)(.log ) b ) (log ) ).(log ) (log ) Solutio : Itgratio. by. parts. w. gt I = (log ).. d.(log ) = (log ). I = (log ). I. + I = d I + I =.(log ) As : b) os +. d = Aos + B, th ot ta aa ) = / ba ) = /8 A ) = / No (os + )os.si Solutio : d os si os.si = d si.os. d si. d os = os = = + 8 As : b)
. + ( )( + ) d If. I = Alog + B ta. th. A & Bis + a), b), ),, Solutio : + d log ta = + + + As : ) 5..(log ) d a)8 (log ) b ). 8(log ) log ). 8(log ) log + + Solutio : Itgratio. by. parts.. (log ) log + = + + + (log ) (log ).. d (log ).8 6.log d = 6 6 (log ).8 (log )... d (log ).8.(log ) + + + + As : b) 8(log ) log