Γιώργος Μαυρίδης-Ολοκληρώματα

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Ασκήσεις εφ' όλης της ύλης στα ολοκληρώματα, από το ανανεωμένο βιβλίο του Γιώργου Μαυρίδη για την Γ΄ Λυκείου. Οι ασκήσεις είναι μια φιλική προσφορά του φίλου και συναδέλφου, Γιώργου Μαυρίδη.

Transcript of Γιώργος Μαυρίδης-Ολοκληρώματα

Page 1: Γιώργος Μαυρίδης-Ολοκληρώματα

9 786188 021495

ISBN: 978-618-80214-9-5

Page 2: Γιώργος Μαυρίδης-Ολοκληρώματα

nȍȨȦ ȰȪȵȦȨ�ȩȨȮ�ȚȯȢȞȪȷȦȞȭȞ�ȭȷȪȚ� ȜȢȚ�ȦȚ ȣȚȭȚȦȨȗȬȞȭȞ�ȭȢȫ�țȚȬȢȣȖȫ�ȖȦȦȨȢȞȫ� ȣȚȢ�ȦȚ�ȞȥțȚȡȶȦȞȭȞ�ȬȭȨȦ�ȭȪȵȩȨ�ȬȣȖȱȠȫ� ȣȚȢ ȞȪȜȚȬȘȚȫ��ȡȚ�ȭȨȦ�ȣȞȪȝȘȬȞȭȞ� ȩȨȤȤȚȩȤȕȬȢȨ�ȚȪȜȵȭȞȪȚ.»

ȁȞȵȝȲȪȨȫ�Ȇ��ȃȚȟȚȦȭȟȗȫ

Page 3: Γιώργος Μαυρίδης-Ολοκληρώματα

ƲƳƱƭƱƥƱƴ

ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ� ȞĮ� ĮʌȠIJİȜȑıȠȣȞ� ȝȑȡȠȢ� IJȠȣȜȐȤȚıIJȠȞ� IJȘȢ�ĮʌȐȞIJȘıȘȢ�ıIJȠȞ�ȚįİȫįȘ�țȩıȝȠ�IJȠȣ��ȥȑȝĮIJȠȢ�ȩʌȠȣ�IJĮ�ʌȐȞIJĮ�įȘȜȫȞȠȣȞ�țĮȚ�įȘȜȫ�ȞȠȞIJĮȚ�ȦȢ�ĮȜȘșȚȞȐ��

ĬĮ�ȝʌȠȡȠȪıĮȞ� IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�ĮʌȠIJİȜȑıȠȣȞ� IJȘȞ�țĮȪıȚȝȘ�ȪȜȘ�Ș�ȠʌȠȓĮ�șĮ�İȞİȡȖȠʌȠȚȠȪıİ�IJȠȞ�ȞĮȡțȦȝȑȞȠ�ȖȓȖĮȞIJĮ�IJȘȢ�ıȣȞİȓįȘıȘȢ��

ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ�� ȦȢ� ȖȜȫııĮ� IJȘȢ� ȜȠȖȚțȒȢ�� ȞĮ� ıȣȞįȡȐȝȠȣȞ�ĮʌȠijĮıȚıIJȚțȐ�IJȚȢ�ıIJȡȚȝȦȖȝȑȞİȢ�įȣȞȐȝİȚȢ�IJȘȢ�ıȪȞİıȘȢ��IJȘȢ�IJĮʌİȚȞȩIJȘIJĮȢ��IJȘȢ�ȖȞȫ�ıȘȢ�țĮȚ�IJȘȢ�ĮȜȒșİȚĮȢ��Įʌȩ�IJĮ�ȝĮȞȚĮıȝȑȞĮ�ıIJȓijȘ�IJȠȣ�©șȑȜȦª��IJȠȣ�©ijĮȓȞȠȝĮȚª��IJȠȣ�©ȑȤȦª��IJȠȣ�©ȖȡȒȖȠȡĮª��IJȠȣ�©țĮIJĮȞĮȜȫȞȦª��IJȠȣ�©İȞIJȣʌȦıȚĮțȠȪª��IJȠȣ�©İȖȫª�

ȅ�țĮșȑȞĮȢ�Įʌ¶�ȩıȠȣȢ�ȕȡȓıțȠȣȞ�ȗȦȠȖȩȞȠ�IJȠ�ȞĮ�ĮıȤȠȜȠȪȞIJĮȚ�ıİ�ȕȐșȠȢ�ȝİ�IJȘ�ȗȦȒ��șĮ�ȑįȚȞİ�IJȘ�įȚțȒ�IJȠȣ�ĮʌȐȞIJȘıȘ�

ǻȪıțȠȜĮ�ʌȐȞIJȦȢ�șĮ�ȣʌȒȡȤĮȞ�įȚĮijȦȞȓİȢ��ıIJȠ�ȩIJȚ�șĮ�ȒIJĮȞ��ĮįȪȞĮIJȠȞ��ĮțȩȝȘ�țĮȚ�ȖȚĮ�IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�İʌȚIJİȜȑıȠȣȞ�ȑȡȖȠ�ıİ�țȐʌȠȚȠ�Įʌȩ�IJĮ�ʌȡȠȘȖȠȪȝİȞĮ��ĮȞ�țȐIJȚ�IJĮ�ĮʌȑțȠʌIJİ�Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȡȠijȩ�IJȠȣȢ��Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȠȣ�ĮʌİȓȡȠȣ�

ǻİȞ�ȞȠȝȓȗȦ�ȩIJȚ�ȤȡİȚȐȗİIJĮȚ�Ș�ıȣȞįȡȠȝȒ�IJȘȢ�țȕĮȞIJȠȝȘȤĮȞȚțȒȢ�� IJȘȢ�İȟİȜȚțIJȚ�țȒȢ�ȕȚȠȜȠȖȓĮȢ�Ȓ�IJȘȢ�șİȦȡȓĮȢ�IJȠȣ�ȤȐȠȣȢ��ȫıIJİ�ȞĮ ȠȡȚıIJİȓ�ȦȢ ȜȠȖȚțȒ�ĮȞĮțȠȜȠȣșȓĮ���Ș�ıȪȞįİıȘ�IJȠȣ�ĮʌİȓȡȠȣ�IJȦȞ�įȣȞĮIJȫȞ�ʌİȡȚʌIJȫıİȦȞ��ȝİ�IJȘȞ�IJȣȤȠįȚȦțIJȚțȠȪ�IJȪʌȠȣ��İʌȚȜȠȖȒ�țȐʌȠȚȦȞ�Įʌȩ�ĮȣIJȑȢ�

Ȃ¶�ĮȣIJȒ� ĮțȡȚȕȫȢ� IJȘȞ� ʌİʌȠȓșȘıȘ� ĮʌȑijȣȖĮ� İʌȚȝİȜȫȢ�� țȐșİ� ȠȜȠțȜȘȡȦIJȚțȠȪ�IJȪʌȠȣ�ȠȝĮįȠʌȠȓȘıȘ�IJȦȞ�ĮıțȒıİȦȞ��ǹȣIJȩ�įȘȜĮįȒ�ʌȠȣ�IJȩıȠ�İȪıIJȠȤĮ�Ƞ�ĮȟȑȤĮıIJȠȢ�įȐıțĮȜȠȢ�ĬİȩįȦȡȠȢ�ȀĮȗĮȞIJȗȒȢ�ȤĮȡĮțIJȒȡȚȗİ�ȦȢ�©ʌİȡȚʌIJȦıȚȠȜȠȖȓĮª�

ǻİȞ�ȑȤİȚ�ȞȩȘȝĮ�ȞĮ�ĮȞĮȗȘIJȒıİȚ�țĮȞİȓȢ�IJȚȢ�ĮȚIJȓİȢ�ĮȣIJȒȢ�IJȘȢ�İȝȝȠȞȒȢ��İȐȞ�įȘ�ȜĮįȒ�ȠijİȓȜİIJĮȚ�ıİ� ȑȜȜİȚȝȝĮ�ȜȠȖȚțȒȢ�Ȓ� İȞIJȚȝȩIJȘIJĮȢ��ȆȐȞIJȦȢ�� ȝȠȣ� İȓȞĮȚ� ȚįȚĮȓIJİȡĮ�įȪıțȠȜȠ�ȞĮ�ĮʌȠijȪȖȦ�IJȠȞ�ʌİȚȡĮıȝȩ�țȐʌȠȚȦȞ�İȞȠȤȜȘIJȚțȫȞ�ĮȜȜȐ�ȝȐȜȜȠȞ�ʌȡĮȖȝĮIJȚ�țȫȞ� ıțȑȥİȦȞ��ȦȢ� ʌȡȠȢ� IJĮ� ĮʌȠIJİȜȑıȝĮIJĮ� ĮȣIJȫȞ� IJȦȞ� ĮȞIJȚȜȒȥİȦȞ� Ȓ� ȓıȦȢ� “ȠȡijĮ�ȞȫȞ” ʌȡĮțIJȚțȫȞ�

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ȀȐșİ�ȥİȣįĮȓıșȘıȘ�țĮȚ�ȝȚĮ�ijȣȜĮțȒ���ȀȐʌȦȢ�ȑIJıȚ�țȚ�İįȫ�Ș�ȥİȣįĮȓıșȘıȘ�IJȘȢ�ıȣıIJȘȝĮIJȚțȒȢ�ȠȡȖȐȞȦıȘȢ�țĮȚ�ȝİȜȑIJȘȢ�ijȣȜĮțȓȗİȚ�IJȠ�įȡȠıİȡȩ�ʌȞİȪȝĮ�IJȦȞ�ȝĮșȘIJȫȞ��ıIJİȡȫȞIJĮȢ�Įʌ¶�ĮȣIJȠȪȢ�IJȘ�ȝĮșȘIJİȓĮ�IJȘȢ�İȟİȡİȪȞȘıȘȢ��IJȘȞ�ĮȞİȝİȜȚȐ�IJȘȢ�ʌİȡȚʌȜȐȞȘ�ıȘȢ�țĮȚ�IJȘ�ȤĮȡȐ�IJȘȢ�ĮȞĮțȐȜȣȥȘȢ��ȈIJİȡȫȞIJĮȢ�ijȣıȚțȐ�ıIJȘ�ıȣȞȑȤİȚĮ�ıIJȠ�ȕĮșȝȩ�ʌȠȣ�IJĮ�İʌȘȡİȐȗȠȣȞ��IJĮ�ȝĮșȘȝĮIJȚțȐ�ĮȜȜȐ�țȣȡȓȦȢ�IJȘȞ�țȠȚȞȦȞȓĮ��Įʌȩ�ȝȣĮȜȐ�İȟİȡİȣȞȘȝȑ�ȞĮ�țĮȚ�ȐȡĮ�İȟİȡİȣȞȘIJȚțȐ��ȝİ� IJİȜȚțȩ�ĮʌȠIJȑȜİıȝĮ�IJȘȞ�ȤȦȡȓȢ�ĮȞIJȓʌĮȜȠ�İʌȚțȡȐIJȘıȘ�IJȘȢ�ȝȓȝȘıȘȢ��IJȘȢ�ijĮȚįȡȩIJȘIJĮȢ�țĮȚ�IJȘȢ�İʌĮȞȐȜȘȥȘȢ��ȅȚ�ĮʌȠįİȓȟİȚȢ�ȩȜȦȞ�IJȦȞ�ʌȡȠȘ�ȖȠȪȝİȞȦȞ��ʌȡȠijĮȞȫȢ�įİȞ�ȤȡİȚȐȗİIJĮȚ�ȞĮ�ĮȞĮȗȘIJȘșȠȪȞ ıİ�ıțȠȞȚıȝȑȞĮ�ȝĮșȘȝĮIJȚțȐ�ȕȚȕȜȓĮ��ǹȞ�IJİȜȚțȐ�ijȠȕȓȗİȚ�IJȩıȠ�ʌȠȜȪ�ȝȚĮ�ȝĮIJȚȐ�ȝȑıĮ�ȝĮȢ��IJȠȣȜȐȤȚıIJȠȞ�IJȫȡĮ�İȓȞĮȚ�ĮȡțİIJȒ�ȝȚĮ�ȝĮIJȚȐ�įȓʌȜĮ�ȝĮȢ�

ĬİııĮȜȠȞȓțȘ��ǿȠȪȞȚȠȢ 2013 īȚȫȡȖȠȢ ȁ��ȂĮȣȡȓįȘȢ

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ƲƧƳƫƧƸƱƮƧƯƣ ƣƘ�ƵƱƮƱƶ

ȆȡȩȜȠȖȠȢ�

ȂȚȖĮįȚțȠȓ�ĮȡȚșȝȠȓ ........................................................................................ 11 � ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 113 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 141 ǵȡȚȠ�–�ȈȣȞȑȤİȚĮ�ıȣȞȐȡIJȘıȘȢ ....................................................................... 151� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 153 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 347 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 391 �ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��ǹǯ�ȂȑȡȠȢ� ............................................................... 401 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 403 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 477 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 501 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 507 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 569

Page 6: Γιώργος Μαυρίδης-Ολοκληρώματα

ƲƧƳƫƧƸƱƮƧƯƣ ƤƘ�ƵƱƮƱƶ

ȆȡȩȜȠȖȠȢ� ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��Ǻǯ�ȂȑȡȠȢ� ............................................................... 11� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 259 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 317 ȅȜȠțȜȘȡȦIJȚțȩȢ�ȁȠȖȚıȝȩȢ ........................................................................... 329 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 331 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 505 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 543 ǼʌĮȞĮȜȘʌIJȚțȐ�ĬȑȝĮIJĮ ................................................................................ 555 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 575 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 636

Page 7: Γιώργος Μαυρίδης-Ολοκληρώματα

486 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ƪƾǍǂ�102�

DzıIJȦ ıȣȞȐȡIJȘıȘ� f : o\ \ Ș�ȠʌȠȓĮ�İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�IJȑIJȠȚĮ��ȫıIJİ� � �f x 0t ȖȚĮ�țȐșİ x�\

țĮȚ

� �� �x u

01f t dt du 2x 2t �µ́

¶ ³ ȖȚĮ�țȐșİ�x�\ .

i) ǹȞ�Ș�ıȣȞȐȡIJȘıȘ�g : o\ \ ȑȤİȚ�IJȪʌȠ

� � � �� �x u

01g x f t dt du 2x 2 � �µ́

¶ ³ ȖȚĮ�țȐșİ x�\ �

ȞĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ� Į� Ș ıȣȞȐȡIJȘıȘ g ʌĮȡȠȣıȚȐȗİȚ�ȠȜȚțȩ�İȜȐȤȚıIJȠ ȕ� Ș�ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ. ii) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�

� �1

0f t dt 2. ³

iii) NĮ� ȕȡİȓIJİ� IJȠ� İȝȕĮįȩ IJȠȣ� ȤȦȡȓȠȣ� ʌȠȣ� ʌİȡȚțȜİȓİIJĮȚ� Įʌȩ� IJȘ� ȖȡĮijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ ıȣȞȐȡIJȘıȘȢ f��IJȠȞ�ȐȟȠȞĮ�IJȦȞ�x� IJȠȞ�ȐȟȠȞĮ�IJȦȞ�y țĮȚ�IJȘȞ�İȣșİȓĮ� x 1.

�ƭǞǔLj� i) Į� ǹʌȩ�IJĮ�įİįȠȝȑȞĮ�IJȠȣ�ʌȡȠȕȜȒȝĮIJȠȢ�ʌȡȠțȪʌIJİȚ�ȩIJȚ

� �g x 0t ȖȚĮ�țȐșİ� x .�\

ǼʌȓıȘȢ,

� � � �1 u

01

g 1 f t dt du 2 2 0.§ · � � ¨ ¸© ¹

´ µ́µ ¶¶

DZȡĮ, � � � �g x g 1t ȖȚĮ�țȐșİ� x .�\

ǻȘȜĮįȒ��Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ ȠȜȚțȩ�İȜȐȤȚıIJȠ�ıIJȠ 0x 1 IJȠ� � �g 1 0.

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ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 487�

ȕ) ǹʌȩ�IJȘ�ıȣȞȑȤİȚĮ� IJȘȢ�ıȣȞȐȡIJȘıȘȢ� � f ıIJȠ�įȚȐıIJȘȝĮ� \ ıȣȝʌİȡĮȓȞȠȣȝİ� IJȘȞ�

ʌĮȡĮȖȦȖȚıȚȝȩIJȘIJĮ��ȐȡĮ�țĮȚ�IJȘ�ıȣȞȑȤİȚĮ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�� � �u

0f t dtµ́

¶ ıIJȠ�ȓįȚȠ�

įȚȐıIJȘȝĮ�� ȚįȚȩIJȘIJĮ�ʌȠȣ�İȓȞĮȚ� ȚțĮȞȒ�ȞĮ�İȖȖȣȘșİȓ� IJȘȞ�ʌĮȡĮȖȦȖȚıȚȝȩIJȘIJĮ� IJȘȢ�

ıȣȞȐȡIJȘıȘȢ � �� �x u

01f t dt duµ́

¶ ³ ıIJȠ� \ . ǼʌȠȝȑȞȦȢ, Ș�ıȣȞȐȡIJȘıȘ�g İȓȞĮȚ�ʌĮ�

ȡĮȖȦȖȓıȚȝȘ�ȝİ�

� � � �� �x u

01g x f t dt du 2x 2

c§ ·c � �¨ ¸© ¹µ́¶ ³

� �� � � � � �x u

01f t dt du 2x 2

c§ · c c � �¨ ¸© ¹µ́¶ ³

� �x

0f t dt 2 �³ ȖȚĮ�țȐșİ�� x .�\

ii) ȈIJȠ�İȡȫIJȘȝĮ� i) ĮʌȠįİȓȟĮȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�g ʌĮȡȠȣıȚȐȗİȚ�ȠȜȚțȩ�İȜȐȤȚıIJȠ�

ıIJȠ� 0x 1, İıȦIJİȡȚțȩ�IJȠȣ�įȚĮıIJȒȝĮIJȠȢ�\ ıIJȠ�ȠʌȠȓȠ�İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��DZȡĮ,

ıȪȝijȦȞĮ�ȝİ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Fermat ȚıȤȪİȚ Ș�ıȤȑıȘ

� �g 1 0c

ǻȘȜĮįȒ,

� �1

0f t dt 2 0� �³ � �

1

0f t dt 2. ³

iii) DzȤȠȣȝİ�

� �f x 0t ȖȚĮ�țȐșİ�� x .�\

ǼʌȠȝȑȞȦȢ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ

� � � �1 1

0 0E f x dx f t dt 2 IJ�ȝ� ³ ³

Page 9: Γιώργος Μαυρίδης-Ολοκληρώματα

488 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ƪƾǍǂ�103�

DzıIJȦ�ıȣȞȐȡIJȘıȘ�� > @I � �� � o \ Ș�ȠʌȠȓĮ�İȓȞĮȚ�ıȣȞİȤȒȢ�ȝİ� � � � �f 0 f 2z țĮȚ IJȑIJȠȚĮ� ȫıIJİ

� � � � � �� �x 2

0

1f x f 2 f t dt3

´µ¶

� ȖȚĮ�țȐșİ� > @[ �� � ��

i) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�� � �f 2 0z .

ii) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ�� > @[ �� � ��

iii) ȃĮ�ȕȡİȓIJİ�IJȠȞ�IJȪʌȠ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�f . iv) ȃĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ�ȖȡĮ�

ijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ

� � � �� � > @2J [ I [ � [ �� � �

IJȠȞ�ȐȟȠȞĮ� x xc țĮȚ�IJȚȢ�İȣșİȓİȢ�� x 1 țĮȚ�� x 2.

ƭǞǔLj�

i) ȊʌȠșȑIJȠȣȝİ� �ĮʌĮȖȦȖȒ� ıİ� ȐIJȠʌȠ�� ȩIJȚ � �f 2 0. ĬȑIJȠȞIJĮȢ� ıIJȘ� įȠșİȓıĮ� ıȤȑıȘ� ȩʌȠȣ�x IJȠ���ȑȤȠȣȝİ

� � � � � �� �0

2

0

1 1f 0 f 2 f t dt 0 0 03 3

� � µ́¶

țĮȚ�ıȣȞİʌȫȢ � � � �f 0 f 2

ʌȠȣ��ȝİ�ȕȐıȘ�IJȘȞ�ȣʌȩșİıȘ��İȓȞĮȚ ȐIJȠʌȠ��ǼʌȠȝȑȞȦȢ � �f 2 0.z

ii) ĬȑIJȠȞIJĮȢ�ıIJȘ�įȠșİȓıĮ�ıȤȑıȘ�ȩʌȠȣ�[�IJȠ���ȑȤȠȣȝİ

� � � � � �� �2 2

0

1f 2 f 2 f t dt3

� µ́¶

Ȓ�ȚıȠįȪȞĮȝĮ

� � � �� �2 2

0

2 f 2 f t dt3

µ́¶

įȘȜĮįȒ������

� � � �� �2 2

0

3f 2 f t dt2

µ́¶

(1)

ȂǼĬȅǻȅȁȅīǿǹ ȆĮȡĮIJȘȡȠȪȝİ�ȩIJȚ

� �� �x 2

0f t dt 0tµ́

¶ ȖȚĮ�țȐșİ� > @x 0, 2�

ȅʌȩIJİ��ȜȩȖȦ�IJȘȢ�įȠșİȓıĮȢ�ȚıȩIJȘIJĮȢ

� � � � � �� �x 2

0

1f x f 2 f t dt3

� µ́¶

ȖȚĮ� ȞĮ� ĮʌȠįİȓȟȠȣȝİ� IJȠ� ȗȘIJȠȪȝİȞȠ�Įȡțİȓ�ȞĮ�ĮʌȠįİȓȟȠȣȝİ�ȩIJȚ � �f 2 0.!

Page 10: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 489�

ǵȝȦȢ��Ș�ıȣȞİȤȒȢ�ıȣȞȐȡIJȘıȘ� � �� �2f t İȓȞĮȚ�ȝȘ�ĮȡȞȘIJȚțȒ�țĮȚ�ȩȤȚ�ʌĮȞIJȠȪ�ȝȘįȑȞ�ıIJȠ�

įȚȐıIJȘȝĮ� > @0, 2 ĮijȠȪ � �� �2f 2 0z . DZȡĮ�

� �� �2 2

0f t dt 0!µ́

ȅʌȩIJİ��Įʌȩ�IJȘ�ıȤȑıȘ�(1) ʌȡȠțȪʌIJİȚ�ȩIJȚ � �f 2 0! . ǼʌȓıȘȢ�

� �� �x 2

0f t dt 0tµ́

¶ ȖȚĮ�țȐșİ�� > @x 0, 2� .

ǼʌȠȝȑȞȦȢ�

� � � � � �� �x 2

0

1f x f 2 f t dt 03

� !µ́¶

ȖȚĮ�țȐșİ�� > @x 0, 2� .

iii) Ǿ� ıȣȞȐȡIJȘıȘ� � �� �2f t İȓȞĮȚ� ıȣȞİȤȒȢ� țĮȚ� ıȣȞİʌȫȢ� Ș� ıȣȞȐȡIJȘıȘ� � �� �

x 2

0f t dtµ́

İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȅʌȩIJİ��Įʌȩ�IJȘ�įȠșİȓıĮ�ıȤȑıȘ�ʌȡȠțȪʌIJİȚ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ�ȝİ�

� � � � � �� �

� � � �� � � �� �

� �� � > @

x 2

0

x 2 2

0

2

1f x f 2 f t dt3

1 f 2 f t dt 0 f x3

f x 0 ȖȚĮ țȐșİ [ �� � �

c§ ·c �¨ ¸© ¹

cc § ·§ · � �¨ ¸ ¨ ¸© ¹ © ¹

! �

µ́¶

µ́¶

ǻȘȜĮįȒ�

� �� �2

f x1

f xc

� � ȖȚĮ�țȐșİ�� > @x 0, 2�

� � � �1 x

f x

c§ · c� �¨ ¸¨ ¸© ¹

ȖȚĮ�țȐșİ�� > @x 0, 2�

� �1 x c

f x� � � ȖȚĮ�țȐșİ��� > @x 0, 2� ���ȝİ�� c�\

� � 1f xx c

� � �

ȖȚĮ�țȐșİ��� > @x 0, 2 .�

Page 11: Γιώργος Μαυρίδης-Ολοκληρώματα

490 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ǵȝȦȢ�� Įʌȩ� IJȘ� įȠșİȓıĮ� ıȤȑıȘ�� șȑIJȠ�

ȞIJĮȢ�ȩʌȠȣ�x IJo ��ȑȤȠȣȝİ

� � � � � �� �

� � � �

� �

0 2

0

1f 0 f 2 f t dt31 1 1 1f 0 f 23 0 c 3 2 c

3 2 c c 6 3c c

2c 6 c 3.

� � �� � � �

� � � � � �

� �

µ́¶

ǼʌȠȝȑȞȦȢ�

� � 1f xx 3

� �

ȖȚĮ�țȐșİ��� > @x 0, 2 .�

iv) H ıȣȞȐȡIJȘıȘ�

� � � �� � > @2g t f t , t 0, 2 �

İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�ȝȘ�ĮȡȞȘIJȚțȒ��ȅʌȩIJİ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ İȓȞĮȚ�ȓıȠ�ȝİ

� � � �� �

� �� �� �

2 2 2

1 122 22

2 211 1

2 2

11

E g t dt f t dt

3 t1 1dt dt dtt 3 3 t 3 t

1 1 1 1 1 1dt 1 IJ�ȝ�3 t 3 t 3 2 3 1 2 2

c� �§ · ¨ ¸� �© ¹ � �

c§ · ª º � � ¨ ¸ « »� � � �© ¹ ¬ ¼

´ ´µ µ¶ ¶

´´ ´µµ µµ µ¶¶ ¶

´µ¶

Ȉȋȅȁǿȅ ǻİȞ�ȣʌȐȡȤİȚ�țȐʌȠȚĮ�ĮȡȤȚțȒ�ıȣȞșȒ�țȘ�ʌȠȣ�șĮ�ȕȠȘșȠȪıİ�ıIJȠȞ�ȣʌȠȜȠȖȚ�ıȝȩ�IJȘȢ�ıIJĮșİȡȐȢ�c. ǻȘȝȚȠȣȡȖȠȪȝİ�İȝİȓȢ� ȜȠȚʌȩȞ� ȝȓĮ� IJȑIJȠȚĮ� ıȣȞșȒțȘ�șȑIJȠȞIJĮȢ�ıIJȘ�įȠșİȓıĮ�ıȤȑıȘ�ȩʌȠȣ x IJȠ� �� țĮIJĮȜȒȖȠȞIJĮȢ� ȑIJıȚ� ıIJȘȞ� Țıȩ�IJȘIJĮ

� � � �1f 0 f 2 .3

Page 12: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 491�

ƪƾǍǂ�104�

ǻȓȞȠȞIJĮȚ�ȠȚ�ıȣȞĮȡIJȒıİȚȢ� I � J � o\ \ ȠȚ�ȠʌȠȓİȢ�İȓȞĮȚ�ıȣȞİȤİȓȢ�țĮȚ�IJȑIJȠȚİȢ��ȫıIJİ

� � � �

x

0

tf x dt 1g t

�´µ¶

ȖȚĮ�țȐșİ�x�\

țĮȚ�

� � � �

x

0

tg x dt 1f t

�´µ¶

ȖȚĮ�țȐșİ� x�\ .

i) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ� x�\

țĮȚ � �g x 0! ȖȚĮ�țȐșİ� x�\ .

ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � � � �f x g x ȖȚĮ�țȐșİ� x�\ .

iii) ȃĮ�ȕȡİȓIJİ�IJȠȞ�IJȪʌȠ�IJȘȢ� ıȣȞȐȡIJȘıȘȢ�f. iv) ȃĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ�ȖȡĮ�

ijȚțȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ

� � � �

x

1

1K [ GW�f t

´µ¶

x�\

IJȠȣȢ�ȐȟȠȞİȢ� x xc �� y yc țĮȚ�IJȘȞ�İȣșİȓĮ� x 1. ƭǞǔLj� i) ǹʌȩ�IJȚȢ�įȠșİȓıİȢ�ıȤȑıİȚȢ�ʌȡȠțȪʌIJİȚ�ȩIJȚ

� �f x 0z ȖȚĮ�țȐșİ� x�\

țĮȚ�� � �g x 0z ȖȚĮ�țȐșİ�� x�\ .

ǻȚĮijȠȡİIJȚțȐ��ȝȓĮ�IJȠȣȜȐȤȚıIJȠȞ�Įʌȩ�IJȚȢ�ıȣȞĮȡIJȒıİȚȢ�� �t

g t țĮȚ�

� �t

f t įİȞ�șĮ�ȠȡȓȗȠ�

ȞIJĮȞ�ıIJȠ� \ țĮȚ�ıȣȞİʌȫȢ��ȝİ�ȕȐıȘ�IJȚȢ�įȠșİȓıİȢ�ıȤȑıİȚȢ��ȝȓĮ�IJȠȣȜȐȤȚıIJȠȞ�Įʌȩ�IJȚȢ�ıȣȞĮȡIJȒıİȚȢ��f țĮȚ��g įİȞ�șĮ�İȓȤİ�ʌİįȓȠ�ȠȡȚıȝȠȪ�IJȠ�\ ��ʌȡȐȖȝĮ�ȐIJȠʌȠ��ǼʌȓıȘȢ��ȠȚ�

Page 13: Γιώργος Μαυρίδης-Ολοκληρώματα

492 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ıȣȞĮȡIJȒıİȚȢ�f țĮȚ� g İȓȞĮȚ�ıȣȞİȤİȓȢ�ıIJȠ�\ țĮȚ�ıȣȞİʌȫȢ�įȚĮIJȘȡȠȪȞ�ʌȡȩıȘȝȠ�ıIJȠ�.\ ȀĮȚ�İʌİȚįȒ

� � � �

0

0

tf 0 dt 1 0 1 1 0g t

� � !´µ¶

țĮȚ � � � �

0

0

tg 0 dt 1 0 1 1 0f t

� � !´µ¶

ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ� � �f x 0! ȖȚĮ�țȐșİ� x�\

țĮȚ � �g x 0! ȖȚĮ�țȐșİ� x�\ .

ii) ȅȚ�ıȣȞĮȡIJȒıİȚȢ � �t

g t țĮȚ�

� �t

f t İȓȞĮȚ�ıȣȞİȤİȓȢ�ıIJȠ� \ . ǼʌȠȝȑȞȦȢ��Įʌȩ�IJȚȢ�įȠșİȓ�

ıİȢ�ıȤȑıİȚȢ�ʌȡȠțȪʌIJİȚ�ȩIJȚ�ȠȚ�ıȣȞĮȡIJȒıİȚȢ� � �f x țĮȚ� � �g x İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝİȢ�ȝİ

� � � � � � � �

x

0

t xf x dt 1g t g x

c§ · cc � ¨ ¸¨ ¸© ¹

´µ¶

ȖȚĮ�țȐșİ�� x�\

țĮȚ

� � � � � � � �

x

0

t xg x dt 1f t f x

c§ · cc � ¨ ¸¨ ¸© ¹

´µ¶

ȖȚĮ�țȐșİ�� x�\ .

ǹʌȩ�IJȚȢ�ʌĮȡĮʌȐȞȦ�ȚıȩIJȘIJİȢ�ȑȤȠȣȝİ

� �� � � � � �

f x xf x f x g xc

ȖȚĮ�țȐșİ�� x�\

țĮȚ � �� � � � � �

g x xg x f x g xc

ȖȚĮ�țȐșİ�� x�\ .

ȅʌȩIJİ,

� �� �

� �� �

f x g xf x g xc c

ȖȚĮ�țȐșİ�� x�\ .

ǻȘȜĮįȒ�

� �� � � �� �ln f x ln g xc c ȖȚĮ�țȐșİ�� x�\ Ȓ�ȚıȠįȪȞĮȝĮ

� � � �ln f x ln g x c � ȖȚĮ�� x�\ ���ȩʌȠȣ� c�\ .

Page 14: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 493�

ĬȑIJȠȞIJĮȢ�ıIJȘȞ�IJİȜİȣIJĮȓĮ�ȚıȩIJȘIJĮ�ȩʌȠȣ�x IJȠ�� ʌĮȓȡȞȠȣȝİ

� � � �ln f 0 ln g 0 c � ln1 ln1 c� � c 0.�

DZȡĮ�

� � � �ln f x ln g x ȖȚĮ�țȐșİ� x�\

țĮȚ�IJİȜȚțȐ

� � � �f x g x ȖȚĮ�țȐșİ� x�\ .

iii) ǹʌȩ�IJȠ�ʌȡȠȘȖȠȪȝİȞȠ�İȡȫIJȘȝĮ�ȑȤȠȣȝİ

� � � � � �x xf x

g x f xc ȖȚĮ�țȐșİ� x�\ .

ǻȘȜĮįȒ,

� � � �2f x f x 2xc ȖȚĮ�țȐșİ� x�\

� �� � � �2 2f x xc c� ȖȚĮ�țȐșİ� x�\

� �2 2f x x c� � ȖȚĮ�țȐșİ� x�\ ��ȩʌȠȣ� c�\ .

ĬȑIJȠȞIJĮȢ�ıIJȘȞ�IJİȜİȣIJĮȓĮ�ıȤȑıȘ�ȩʌȠȣ�x IJȠ���ʌĮȓȡȞȠȣȝİ

� �2 2f 0 0 c � � c 1.

ǼʌȠȝȑȞȦȢ�

� �2 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .

ȀĮȚ�İʌİȚįȒ

� �f x 0! ȖȚĮ�țȐșİ� x�\ ,

ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ

� � 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .

Page 15: Γιώργος Μαυρίδης-Ολοκληρώματα

494 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

iv) ȆĮȡĮIJȘȡȠȪȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ � �1

f t İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ� .\ ǼʌȠȝȑȞȦȢ��Ș�ıȣȞȐȡ�

IJȘıȘ

� � � �

x

1

1h x dtf t

´µ¶

İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȐȡĮ�țĮȚ�ıȣȞİȤȒȢ���ıIJȠ�\ ȝİ

� � � � 2

1 1h x 0f x x 1

c !�

ȖȚĮ�țȐșİ� x�\ .

DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ�h İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ��ȀĮȚ�İʌİȚįȒ

� � � �

1

1

1h 1 dt 0f t

´µ¶

,

ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ

� � � � � �h x 0 h x h 1 x 1� � � � �

țĮȚ

� � � � � �h x 0 h x h 1 x 1.! � ! � !

ǼʌȠȝȑȞȦȢ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ

� � � � � � � �1 1 1

0 00E h x dx h x dx x h x dxc � �´ ´ ´µ µµ ¶ ¶¶

� � � � � � � � � �111

0 20 0

1xh x x h x dx 1h 1 0h 0 x dxx 1

c � � � � � �ª º¬ ¼�

´µ́ µ¶ ¶

� �1 1 1

2 2

2 000

2x0 0 dx x 1 dx x 12 x 1

c ª º � � � �¬ ¼�

´ ´µµ ¶¶

2 21 1 0 1 2 1 � � � � IJ�ȝ�

Page 16: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 495�

ƪƾǍǂ�105�

DzıIJȦ�ıȣȞȐȡIJȘıȘ� f : o\ \ IJȑIJȠȚĮ��ȫıIJİ � � � �1f x f x

t 2x 12

1

e dt e et

� �´µµ¶

ȖȚĮ�țȐșİ��x�\ .

i) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ � �f x 0! ȖȚĮ�țȐșİ��x�\ .

ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�

� � 2f x x 1 � ȖȚĮ�țȐșİ� x�\ .

iii) AȞ�F İȓȞĮȚ�Ș�ʌĮȡȐȖȠȣıĮ�IJȘȢ�f ıIJȠ� \ ȖȚĮ�IJȘȞ�ȠʌȠȓĮ�ȚıȤȪİȚ� � �) � �� ȞĮ�ȣʌȠȜȠȖȓıİIJİ� IJȠ� İȝȕĮįȩ�IJȠȣ� ȤȦȡȓȠȣ�ʌȠȣ�ʌİȡȚțȜİȓİIJĮȚ�Įʌȩ�IJȘ� F& � IJȠȞ�ȐȟȠȞĮ�x xc țĮȚ�IJȚȢ�İȣșİȓİȢ� x 0 țĮȚ��x 1.

ƭǞǔLj��

i) Ǿ�ıȣȞȐȡIJȘıȘ�

1t

2et

ȑȤİȚ�ʌİįȓȠ�ȠȡȚıȝȠȪ�IJȠ�ıȪȞȠȜȠ� *\ țĮȚ�İȓȞĮȚ�ıȣȞİȤȒȢ�ı¶�ĮȣIJȩ�

ȦȢ� ĮʌȠIJȑȜİıȝĮ� ʌȡȐȟİȦȞ� ȝİIJĮȟȪ� ıȣȞİȤȫȞ� ıȣȞĮȡIJȒıİȦȞ�� ǼʌȠȝȑȞȦȢ�� Ș� ʌĮȡȐ�ıIJĮıȘ

� � 1f xt

21

e dtt

´µµ¶

ȠȡȓȗİIJĮȚ�ȖȚĮ�İțİȓȞĮ�IJĮ�x IJȑIJȠȚĮ��ȫıIJİ�IJĮ�ȩȡȚĮ�ȠȜȠțȜȒȡȦıȘȢ���țĮȚ� � �f x ȞĮ�ĮȞȒ�

țȠȣȞ�ıIJȠ�ȓįȚȠ�įȚȐıIJȘȝĮ�IJȠȣ� *.\ ǻȘȜĮįȒ��ȖȚĮ�İțİȓȞĮ�IJĮ�x IJȑIJȠȚĮ��ȫıIJİ�

� � � �f x 0, .� �f

ǵȝȦȢ��ıȪȝijȦȞĮ�ȝİ�IJȘ�įȠșİȓıĮ�ıȤȑıȘ��Ș�İȞ�ȜȩȖȦ�ʌĮȡȐıIJĮıȘ�ȠȡȓȗİIJĮȚ�ȖȚĮ�țȐșİ�x .�\ ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ

� �f x 0! ȖȚĮ�țȐșİ�� x .�\

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496 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ii) Ǿ�įȠșİȓıĮ�ıȤȑıȘ�ȖȡȐijİIJĮȚ

� �� �2

f x f x1t x 1

21

1e dt e et

�§ ·� � � �¨ ¸© ¹

´µ¶

ȖȚĮ�țȐșİ�� x�\

� �� �2

f x f x1t x 1

1

1e dt e et

�c§ ·� � � �¨ ¸

© ¹

´µ¶

ȖȚĮ�țȐșİ�� x�\

� �� �2

f x f x1t x 1

1

e dt e e�c§ ·

� �¨ ¸© ¹

´µµ¶

ȖȚĮ�țȐșİ� x�\

� � � �2

f x f x1t x 1

1

e e e�ª º

� �« »¬ ¼

ȖȚĮ�țȐșİ�� x�\

� �� �2

f x1 1f x 1 x 1e e e e�� � � ȖȚĮ�țȐșİ�� x�\

� �� �2

f x1f x x 1e e �� ȖȚĮ�țȐșİ�� x�\

� �� �

2

f x1f x x 1

� �

ȖȚĮ�țȐșİ�� x�\

� �� �2 2f x x 1� � ȖȚĮ�țȐșİ�� x�\

ȀĮȚ�İʌİȚįȒ

� �f x 0! ȖȚĮ�țȐșİ�� x�\

ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ

� � 2f x x 1 � ȖȚĮ�țȐșİ�� x .�\

iii) Ǿ�ıȣȞȐȡIJȘıȘ�F İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ��ȐȡĮ�țĮȚ�ıȣȞİȤȒȢ�ıIJȠ� .\ ǼʌȠȝȑȞȦȢ��IJȠ�ȗȘ�

IJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ� � �1

0E F x dx µ́

¶. ǼʌȓıȘȢ�

� � � � 2F x f x x 1 0c � ! ȖȚĮ�țȐșİ�� x .�\

DZȡĮ��Ș�ıȣȞȐȡIJȘıȘ�F İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�țĮȚ�ıȣȞİʌȫȢ

� � � � � �F x F 1 0 ȖȚĮ țȐșİ [ ��� � �f .

Page 18: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 497�

DzȤȠȣȝİ�ȜȠȚʌȠȞ

� �F x 0d ȖȚĮ�țȐșİ�� > @x 0,1� .

ȅʌȩIJİ�

� � � �� �1 1

0 0E F x dx F x dx �´ ´µ µ

¶ ¶

� � � � � � � � � �1 11

00 0x F x dx xF x x F x dxc c � � � �ª º¬ ¼

´ ´µ µ¶ ¶

� � � � � �1

01 F 1 0 F 0 xf x dx � � � � � µ́

12

0x x 1dx �µ́

¶.

īȚĮ�IJȠȞ�ȣʌȠȜȠȖȚıȝȩ�IJȠȣ�IJİȜİȣIJĮȓȠȣ�ȠȜȠțȜȘȡȫȝĮIJȠȢ�șȑIJȠȣȝİ

2u x 1 � .

ȅʌȩIJİ 2 2u x 1 �

țĮȚ�ıȣȞİʌȫȢ

� � � �2 2u du x 1 dx 2udu 2xdx udu xdx.c c � � �

īȚĮ� x 0 ȑȤȠȣȝİ� u 1

īȚĮ� x 1 ȑȤȠȣȝİ� u 2.

ǼʌȠȝȑȞȦȢ�

1 1

2 2

0 0x x 1dx x 1 xdx� �´ ´µ µ

¶ ¶

2

1uu du µ́

22

1u du µ́

� �323 3

1

2u 13 3 3

ª º �« »¬ ¼

2 2 1.3�

ǻȘȜĮįȒ��IJȠ�ȗȘIJȠȪȝİȞȠ�İȝȕĮįȩ�İȓȞĮȚ

2 2 1E IJ�ȝ�3�

Page 19: Γιώργος Μαυρίδης-Ολοκληρώματα

498 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ƪƾǍǂ���6�

DzıIJȦ�ıȣȞȐȡIJȘıȘ� f : o\ \ Ș�ȠʌȠȓĮ�İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�IJȑIJȠȚĮ��ȫıIJİ: z � � � �f 1 f 1� � z � �f x 0z ȖȚĮ�țȐșİ�� x *��

z � � � �x

1

f x1 2 f t dt

x� ³ ȖȚĮ�țȐșİ�� �x * .�

i) NĮ�ȕȡİȓIJİ�IJȘȞ�IJȚȝȒ� � �f 0 țĮȚ�IJȠ�ʌȡȩıȘȝȠ�IJȘȢ� f.

ii) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ� � � 2x 1f x xe � ȖȚĮ�țȐșİ� �x� . iii) NĮ�ȕȡİȓIJİ�IJĮ țȠȚȞȐ�ıȘȝİȓĮ�IJȘȢ� fC ȝİ�IJȘȞ�İȣșİȓĮ� y x. iv) NĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ� f ĮȞIJȚıIJȡȑijİIJĮȚ�țĮȚ� ȞĮ�ȣʌȠȜȠȖȓıİIJİ�

IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ʌȠȣ�ıȤȘȝĮIJȓȗȠȣȞ�ȠȚ�ȖȡĮijȚțȑȢ�ʌĮȡĮıIJȐıİȚȢ�IJȦȞ�ıȣȞĮȡIJȒıİȦȞ�f țĮȚ�� 1f .�

ƭǞǔLj�

i) Aʌȩ�IJȘ�ıȤȑıȘ

� � � �x

1

f x1 2 f t dt

x� µ́

¶ ȖȚĮ�țȐșİ�� x *�� (1)

ʌĮȓȡȞȠȣȝİ

� � � �x

1f x x 2x f t dt � µ́

¶ ȖȚĮ�țȐșİ�� x *.�� (2)

Ǿ�ıȣȞȐȡIJȘıȘ� � �f t İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ� ,� ȠʌȩIJİ�Ș�ıȣȞȐȡIJȘıȘ� � �x

1f t dtµ́

¶ İȓȞĮȚ�

ʌĮȡĮȖȦȖȓıȚȝȘ� ȐȡĮ� țĮȚ� ıȣȞİȤȒȢ� ıIJȠ� ȓįȚȠ� įȚȐıIJȘȝĮ�� OʌȩIJİ�� Įʌȩ� IJȘ� ıȤȑıȘ� (2) ʌĮȓȡȞȠȣȝİ

� � � � � �x 0

x 0 x 0 x 0 x 0 1 1limf x lim x lim 2x lim f t dt 0 0 f t dt 0o o o o

� � � � ´ ´µ µ¶ ¶

țĮȚ�ȜȩȖȦ�IJȘȢ�ıȣȞȑȤİȚĮȢ�IJȘȢ�f � � � �x 0

f 0 limf x 0.o

H ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ıȣȞİȤȒȢ�

țĮȚ�įİȞ�ȝȘįİȞȓȗİIJĮȚ�ıİ�țĮȞȑȞĮ�ıȘȝİȓȠ�IJȠȣ� *.� ǼʌȠȝȑȞȦȢ��Ș�f įȚĮIJȘȡİȓ�ʌȡȩıȘ�ȝȠ�ıİ�țĮșȑȞĮ�Įʌȩ�IJĮ�įȚĮıIJȒȝĮIJĮ� � �, 0�f țĮȚ� � �0, .� f ǼʌȓıȘȢ, Įʌȩ�IJȘ�ıȤȑıȘ�(2) ȑȤȠȣȝİ

� � � �1

1f 1 1 2 f t dt 1 0 1 0 � � !µ́

țĮȚ�ıȣȞİʌȫȢ � � � �f 1 f 1 1 0� � � � .

Page 20: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 499�

ǹʌȩ�IJĮ�ʌĮȡĮʌȐȞȦ�ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ � �f x 0� ȖȚĮ�țȐșİ�� � �x , 0� �f

țĮȚ � �f x 0! ȖȚĮ�țȐșİ�� � �x 0, .� �f

ii) Aʌȩ�IJȘ�ıȤȑıȘ�(2) țĮȚ�ȝİ�ȕȐıȘ�IJȘȞ�ʌĮȡĮȖȦȖȚıȚȝȩIJȘIJĮ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ� � �x

1f t dtµ́

ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ʌĮȡĮȖȦȖȓıȚȝȘ�ȝİ�

� � � � � � � � � � � �x x x

1 1 1f x x 2x f t dt 1 2x f t dt 2x f t dt

c c§ · § ·c cc � � �¨ ¸ ¨ ¸© ¹ © ¹

´ ´ ´µ µ µ¶ ¶ ¶

� � � �� � � � � �

x

1

f x1 2 f t dt 2xf x 2xf x

x � � �µ́

1

� �1 2x f xx

§ · �¨ ¸© ¹

ȖȚĮ țȐșİ [ �� .

ǼʌȠȝȑȞȦȢ�� � �� � � �

f x 1 2x ȖȚĮ țȐșİ [ ĮijȠȪ I [ � ȖȚĮ țȐșİ [f x x

c � � z �� � �

Ȓ�ȚıȠįȪȞĮȝĮ

� �� � � �2ln f x ln x x ȖȚĮ țȐșİ [ � c c � �� (3)

z ȈIJȠ�įȚȐıIJȘȝĮ� � �, 0�f İȓȞĮȚ� x 0� țĮȚ�� � �f x 0� ��ȠʌȩIJİ�Ș�ıȤȑıȘ�(3) ȖȓȞİIJĮȚ

� �� �� � � �� �2ln f x ln x xc c� � � ȖȚĮ�țȐșİ�� x 0�

įȘȜĮįȒ � �� � � � 2

1ln f x ln x x c� � � � ȖȚĮ�țȐșİ�� x 0� ȝİ� 1c ��

īȚĮ� x 1 � ʌĮȓȡȞȠȣȝİ

� �� � 1 1 1ln f 1 ln1 1 c ln1 ln1 1 c c 1.� � � � � � � � �

ȅʌȩIJİ,

� �� � � � � �

� �� � � � � �

� �� � � � � �

� � � �

� � � �

2

2

x 1

2x 1

2x 1

2x 1

ln f x ln x x 1 ȖȚĮ țȐșİ [ ��

ln f x ln x ln e ȖȚĮ țȐșİ [ ��

ln f x ln xe ȖȚĮ țȐșİ [ ��

f x xe ȖȚĮ țȐșİ [ ��

f x xe ȖȚĮ țȐșİ [ �� �

� � � � � �f

� � � � � �f

� � � � �f

� � � � �f

� � �f

Page 21: Γιώργος Μαυρίδης-Ολοκληρώματα

500 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

z ȈIJȠ�įȚȐıIJȘȝĮ� � �0, � f İȓȞĮȚ� x 0! țĮȚ� � �f x 0! ��ȠʌȩIJİ�Ș�ıȤȑıȘ�(3) ȖȓȞİIJĮȚ

� �� � � �2ln f x ln x x ȖȚĮ țȐșİ [ �cc � !

įȘȜĮįȒ � � 2

2ln f x ln x x c � � ȖȚĮ�țȐșİ�� x 0! ȝİ�� 2c .��

īȚĮ�� x 1 ʌĮȓȡȞȠȣȝİ � � 2 2 2ln f 1 ln1 1 c ln1 ln1 1 c c 1. � � � � � � �

ǼʌȠȝȑȞȦȢ�

� � � �

� � � �

� � � � � �

� � � �

2

2

2x 1

x 1

2x 1

ln f x ln x x 1 ȖȚĮ țȐșİ [ ��

ln f x ln x ln e ȖȚĮ țȐșİ [ ��

ln f x ln xe ȖȚĮ țȐșİ [ ��

f x xe ȖȚĮ țȐșİ [ �� �

� � � �f

� � � �f

� � �f

� � �f

z ȅ�ʌĮȡĮʌȐȞȦ�IJȪʌȠȢ�ȚıȤȪİȚ�țĮȚ�ȖȚĮ� x 0 ĮijȠȪ�įȓȞİȚ� � �f 0 0 .

DZȡĮ�

� � 2x 1f x xe � ȖȚĮ�țȐșİ�� x .�\

iii) Ǿ�İȟȓıȦıȘ� � �f x x ȚıȠįȪȞĮȝĮ�ȖȡȐijİIJĮȚ

� �2 2x 1 x 1

2x 1

2

xe x 0 x e 1 0

x 0 Ȓ H �x 0 Ȓ [ � �x 0 Ȓ [ � Ȓ [ ��

� �

� � �

� � � �

DZȡĮ��IJĮ�țȠȚȞȐ�ıȘȝİȓĮ�IJȘȢ� fC ȝİ�IJȘȞ�İȣșİȓĮ� y x İȓȞĮȚ�IJĮ�ıȘȝİȓĮ� � �A 1, 1� � ,

� �O 0, 0 țĮȚ�� � �B 1,1 .

iv) DzȤȠȣȝİ

� � � � � � � �2 2 2x 1 x 1 x 1 2

2 2x 1 2 x 1

f x xe x e x e x 1

e 2x e 0 ȖȚĮ țȐșİ [ �

� � �

� �

c ccc � � � � �

� ! ��

DZȡĮ�� Ș� ıȣȞȐȡIJȘıȘ� � f İȓȞĮȚ� ȖȞȘıȓȦȢ� ĮȪȟȠȣıĮ� țĮȚ� ıȣȞİʌȫȢ� 1 1� �� įȘȜĮįȒ� ĮȞIJȚ�ıIJȡȑȥȚȝȘ���ȁȩȖȦ�IJȘȢ�ıȣȝȝİIJȡȓĮȢ�IJȦȞ� fC țĮȚ� 1f

C � ȦȢ�ʌȡȠȢ�IJȘȞ�İȣșİȓĮ� y x Ș�

İȣșİȓĮ�ĮȣIJȒ�ȤȦȡȓȗİȚ�IJȠ�ȤȦȡȓȠ�ȍ��IJȠȣ�ȠʌȠȓȠȣ�IJȠ�İȝȕĮįȩ�ȗȘIJİȓIJĮȚ��ıİ�įȪȠ�ȤȦȡȓĮ� 1ȍ țĮȚ� 2ȍ ʌȠȣ�İȓȞĮȚ�ȓıĮ��ȐȡĮ�țĮȚ�ȚıİȝȕĮįȚțȐ��

Page 22: Γιώργος Μαυρίδης-Ολοκληρώματα

ƱnjǐNjnjLjǒǚǕNJNjǝǓ�ƭǐDŽNJǔǍǝǓ� � 501�

ǹȞ� 1ȍ İȓȞĮȚ�IJȠ�ȤȦȡȓȠ�ʌȠȣ�ıȤȘȝĮIJȓȗȠȣȞ�Ș� fC țĮȚ�Ș��İȣșİȓĮ� y x , IJȩIJİ�ȑȤȠȣȝİ

� � � �1

1 1E ȍ I [ [ G[�

� �³

ȉȠ�ʌȡȩıȘȝȠ�IJȘȢ�įȚĮijȠȡȐȢ

� � � �2x 1f x x x e 1�� �

ıIJȠ�įȚȐıIJȘȝĮ� > @1, 1� ijĮȓȞİIJĮȚ�ıIJȠȞ�ʌĮȡĮțȐIJȦ�ʌȓȞĮțĮ��

ǼʌȠȝȑȞȦȢ�

� � � � � �0 1

11 0

E ȍ I [ [ G[ I [ [ G[�

� � �´ ´µ µ¶ ¶

� �� � � �� �

0 1

1 00 10 12 2x 1 x 1

1 001

f x x dx x f x dx

xe dx xdx xdx xe dx

� �

� �

� � �

� � �

´ ´µ µ¶ ¶

´ ´´ ´µ µµµ¶ ¶¶¶

2

0 10 12 2 2x 1 x 1

1 01 0

1 x x 1e e2 2 2 2

� �

� �

ª º ª ºª º ª º � � �« » « »« » « »¬ ¼ ¬ ¼¬ ¼ ¬ ¼

1 1 1 1 1 10 02e 2 2 2 2 2e

� � � � � � �

1 IJ�ȝ�e

DZȡĮ��IJȠ�İȝȕĮįȩ�IJȠȣ�ȤȦȡȓȠȣ�ȍ�İȓȞĮȚ

� � � �12Ǽ ȍ �Ǽ ȍ IJ�ȝ�e

x 1� 0 1

x � � 2x 1e 1� � � �

� �f x x� � �

0

0 0

0 0 0

Page 23: Γιώργος Μαυρίδης-Ολοκληρώματα

502 ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

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DzıIJȦ�ȝȚȖĮįȚțȩȢ�ĮȡȚșȝȩȢ�z IJȑIJȠȚȠȢ��ȫıIJİ

� �z 2 țĮȚ P ] � , t

i) ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ƞ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�IJȘȢ�İȚțȩȞĮȢ�IJȠȣ�z ıIJȠ ȝȚȖĮįȚ�

țȩ�İʌȓʌİįȠ�İȓȞĮȚ�Ș�ȖȡĮijȚțȒ�ʌĮȡȐıIJĮıȘ�ȝȚĮȢ�ıȣȞȐȡIJȘıȘȢ�f ��ȆȠȚȠȢ�İȓ�

ȞĮȚ�Ƞ�IJȪʌȠȢ�IJȘȢ�f ;

ii) NĮ�ȣʌȠȜȠȖȓıİIJİ�IJȠ�ȠȜȠțȜȒȡȦȝĮ� � �2

2f x dx .

´µµ¶�

ƭǞǔLj� i) DzıIJȦ z x yi � țĮȚ�� � �M x, y Ș�İȚțȩȞĮ�IJȠȣ�ıIJȠ�ȝȚȖĮįȚțȩ�İʌȓʌİįȠ��DzȤȠȣȝİ�

z 2 țĮȚ���� � �Im z 0t .

ǻȘȜĮįȒ�

> @

2 2

2 2

2 2

2 2

2

x y 2 țĮȚ \ �

x y 4 țĮȚ \ �

y 4 x țĮȚ \ �

y 4 x țĮȚ � [ �

y 4 x țĮȚ [ �� � �

� t

� � t

� � t

� � � t

� � � �

DZȡĮ��Ƞ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�IJȘȢ�İȚțȩȞĮȢ�IJȠȣ�z ıIJȠ�ȝȚȖĮįȚțȩ�İʌȓʌİįȠ�İȓȞĮȚ�Ș�ȖȡĮijȚ�

țȒ�ʌĮȡȐıIJĮıȘ�IJȘȢ�ıȣȞȐȡIJȘıȘȢ�

� � > @2f x 4 x , x 2, 2 � � � .

Page 24: Γιώργος Μαυρίδης-Ολοκληρώματα

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ii) DzȤȠȣȝİ

� � 2f x 4 x 0 � t ȖȚĮ�țȐșİ�� > @x 2, 2� � .

ȅʌȩIJİ�� IJȠ� ȗȘIJȠȪȝİȞȠ�ȠȜȠțȜȒȡȦȝĮ�İȓȞĮȚ� ȓıȠ�ȝİ�IJȠ�İȝȕĮįȩ � �E ȍ IJȠȣ� ȤȦȡȓȠȣ� ʌȠȣ� ʌİȡȚ�

țȜİȓİIJĮȚ� Įʌȩ� IJȘ� fC țĮȚ� IJȠȞ� ȐȟȠȞĮ� x x.c ǵȝȦȢ��Įʌȩ�IJȚȢ�įȠșİȓıİȢ�ıȣȞșȒțİȢ

z 2 țĮȚ���� � �Im z 0t

ʌȡȠțȪʌIJİȚ�ȩIJȚ�Ƞ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�IJȘȢ�İȚțȩ�ȞĮȢ�IJȠȣ�z��įȘȜĮįȒ�Ș� fC İȓȞĮȚ�IJȠ�ıȪȞȠȜȠ�IJȦȞ�

ıȘȝİȓȦȞ� IJȠȣ� țȪțȜȠȣ� � �c ȝİ� țȑȞIJȡȠ� IJȠ� ıȘ�

ȝİȓȠ� � �O 0, 0 țĮȚ�ĮțIJȓȞĮ� ȡ � ʌȠȣ�ȑȤȠȣȞ�ȝȘ�

ĮȡȞȘIJȚțȒ� IJİIJĮȖȝȑȞȘ�� � DZȡĮ�� Ș� fC İȓȞĮȚ� ȑȞĮ�

ȘȝȚțȪțȜȚȠ�IJȠȣ� � �c țĮȚ�İʌȠȝȑȞȦȢ

� � 2 21 1E ȍ ʌȡ ʌ � �ʌ�2 2

ȈȣȝʌİȡĮȓȞȠȣȝİ�ȜȠȚʌȩȞ�ȩIJȚ

� �2

2f x dx 2ʌ�

� µ́

ȆǹȇǹȉǾȇǾȈǾ ȅ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�IJȘȢ�İȚțȩȞĮȢ�IJȠȣ� z ȩʌȦȢ� ĮʌȠįİȓȟĮȝİ� İȓȞĮȚ� Ș�

fC ��ǵȝȦȢ��Įʌȩ�IJȚȢ�įȠșİȓıİȢ�ıȤȑ�

ıİȚȢ�ȖȓȞİIJĮȚ�ȐȝİıĮ�ĮȞIJȚȜȘʌIJȩ�ȩIJȚ�ĮȣIJȩȢ�Ƞ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�İȓȞĮȚ�ȑȞĮ� ȘȝȚțȪțȜȚȠ�� Ǿ� įȚĮıIJĮȪȡȦıȘ�IJȦȞ� ʌĮȡĮʌȐȞȦ� ʌȜȘȡȠijȠȡȚȫȞ�ȖİȞȞȐ� ȑȞĮ� ȩȝȠȡijȠ� ıȣȝʌȑȡĮıȝĮ��ȉȠ�ȗȘIJȠȪȝİȞȠ�ȠȜȠțȜȒȡȦȝĮ�İȓȞĮȚ�ȓıȠ� ȝİ� IJȠ� İȝȕĮįȩ� İȞȩȢ� ȘȝȚțȣ�țȜȓȠȣ��

O x

y

fC

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