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òîÜ–þa@Þaë‡Ûa@ :متهيد

الدوال من فإننا نستطيع أن نأيت بعدد غري منتهيذس يساوي ) ميلها ( لو أردنا إجياد الدالة اليت مشتقتها األوىل : ولتوضيح ذلك، يكفي أن نالحظ أنه إذا كان لدينا ذس= اليت ميلها

٥ -@ س= } س{د ذ + @ س= } س{د ا ي ححيث ا + @س= } س{ د @ س= } س{د

لكل دالة من الدوال السابقة ،ذ س = } س{د فإنه وبال شك أننا سنالحظ أن املشتقة األوىل : ننا ميكن أن نعمم الصورة العامة لكل من الدوال السابقة على الصورة وبالتايل فإ

} ذس =}س{عائلة املنحنيات للدوال اليت ميلها د{وتسمى هذه الدالة ي ح ، ث حيث ث +@ س= }س{ د .}}س{ دهي دالة أصلية للدالة} س{د نقول أن { أو} }س{ د= م معادلة جمموعة املنحنيات اليت ميلها {أو

::::مالحظة مالحظة مالحظة مالحظة

لعدم معرفتنا باحلد الثابت فإننا نضع عنه رمز يدل عليه ألننا ال نستطيع أن حندد الدالة األصلية بالضبط وهذا احلد ميكن أن يساوي الصفر أو عدد موجب أو عدد سالب ونرمز له بالرمز ث

:تعريف oãb×@aˆg ح خ فففف الدالة د معرفة على الفترة ، æhÏ كل دالة ل حتقق العالقة : فففف للدالة د على}معكوس املشتقة {أو تسمى دالة أصلية أو تكامل فففف ي سلكل }س{د=}س{ل

���و��@@@@@@@@@ : : : :مترينمترينمترينمترين ������� ٠ي ح ����#��س� �ZE۲سFد@@@@����א���دא��

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٠ لكل مشتقة يوجد عدد ال ائي من الدوال األصلية ~١

ء س}س{دت له بالرمز وسنرمزسبالنسبة لـ } س{دأوجد التكامل لـ = }س{د أوجد الدالة األصلية للدالة ~۲ ٠بالنسبة لـ س ) س(ل دوتقرأ تكام

}س{د= } س{ لؤئ} ث ثابت {ث+ } س{ل= ءس }س{دت ~ ������٣ ثابت التكامل: ث ، ) س(الدالة األصلية للدالة د: حيث ل

:نظرية ����

aˆgoãb×@ب ا (على الفترة صفرا )= س( الدالة د ، (æhÏ ب ا[ د تكون ثابتة يف الفترة ، [

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òÛa‡Ûa@åØnÛ دالتني أصليتني للدالة د على ) س(٢ل ، ) س(١ إذا كانت لفففف د معرفة على الفترة ٠ ففففي ث لكل س ) = س(٢ ل –) س(١ ل : @æbÏ ففففالفترة

�����א��} س{٢ل ،} س{١ ل9ذא�078 : : : :ممممثثثثاااالللل ������� و078@ ذ ،+س٣= } س{د دא��

.. = } س{ه @ فإن }س{١ ل -}س{٢ ل =} س{ ه

٥ ~د ٣ ~ج ۲ ~ب صفر ~ ا..........................................................................................................................................................................................................................................................................

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::::مالحظات مالحظات مالحظات مالحظات �����������

)س(د) = س(ل: فان ) س(دالة أصلية للدالة د) س( إذا كانت ل~١

أو اختصارا الدوال ،)س ( ث الصورة العامة للدالة األصلية د ) + س ( تسمى العبارة ل @@@@@@@@@@@@~۲ ٠أو معادلة جمموعة املنحنيات ) س( د األصلية للدالة

) يف كتاب الطالب ٧٣صفحة ( ليس من الضروري أن يكون لكل دالة معطاة دالة أصلية ~٣

: : : : مثال مثال مثال مثال ــــــ) = س( د<=�دא�������������א�� :@:س :- :١[ �=�Eس3�F>��7ن�א��א�����ل���

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Wتساوي �Eس�Fدهي دالة أصلية للدالة %}س@جتا+ س @جا{= } س{ل الدالةالدالةالدالةالدالة ::::تدريب تدريب تدريب تدريب }س@جا -س @ جتا{٥~ د ١~ ج }$ س@جتا+ س @ جا{٥~ ب صفر~ ا

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:@:س :- :١[ �

س - �

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Ûa@‡ÇaìÓ@�ÈiÝßbØn@

: قواعد التكامل أ أهم

ال تنس تكامل الدالة الثابتة دالة من الدرجة األوىل ث+ اس = اء س ت ~١

ن؛!؛= ء س ن س ت ~۲ س١؛ = ن ١-آل ، ن ث! + + ن؛

خاص التكامل غري احملدد

ث+ } س{د= ءس }س{د ت ~١

ءس }س{دت ا = ءس }س{ادت ~ ۲

ءس}س{رت _ ءس } س{دت = ءس }س{ر} _ س{دأت ~ ٣ ميكن توزيع التكامل على اجلمع والطرح

: ة على قواعد التكاملمثالأ

:أوجد التكامالت التالية ءس ٣ت أل١

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س ءس ٨ت أل۲

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؛ بب؛ا؛ +ب ؛=ا؛ بب+ ١

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ءس} ٧ -#س ٤+ @ س٩{ت أل٣

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٠ ١)=٠(لللل 9ذא�078، ٣-س۲)= س(د �و���א��א���א*��������א�� ::::مثال مثال مثال مثال

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ءسس[ ت :�و���א��?�/�א���=: مثال مثال مثال مثال أل١

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îÛbnÛa@ÝßbØnÛa@‡uëc@ò@@@@@@@Z: تدريب تدريب تدريب تدريب

ءس )٨ – س + #س٥( ت ~٤ ءس٣[ ت ~٣ ءس& _س ت ~۲ ءس:#س[ ت ~١

م؛نن }س{ = : م:س[ ن

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:ÝßbØnÛa@‡ÇaìÓ: تابع

: : : :تكامل حاصل ضرب دالتني أو أكثر ونأخذ فيها الفكرتني التاليتنيتكامل حاصل ضرب دالتني أو أكثر ونأخذ فيها الفكرتني التاليتنيتكامل حاصل ضرب دالتني أو أكثر ونأخذ فيها الفكرتني التاليتنيتكامل حاصل ضرب دالتني أو أكثر ونأخذ فيها الفكرتني التاليتني : : : : أأأأووووالالالال

:ب الدوال يف بعضها ونكاملنضر أل١

@@òîmŁa@pýßbØnÛa@‡uëc@Z@

ءس} ٤ –س ٥} { ٣+ @ س{ت ~١

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ءس}@ ۲ -@ س{ ت ~۲ ..........................................................................................................................................................................................................................................................................

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ءس} ٨+ @ س٣{س ت أوجد ::::تدريب تدريب تدريب تدريب

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ال ميكن توزيع التكامل على الضرب

٦٧

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:شتقتها شتقتها شتقتها شتقتها مممم على صورة دالة يف على صورة دالة يف على صورة دالة يف على صورة دالة يف التكاملالتكاملالتكاملالتكامليكون يكون يكون يكون أل۲

���������������������(?@8�,Bدא�/��9ذא�0ن�8?-�C�D��E��دא��F�GH�E��/��I(Jدא���دא(/�,>�������Eس��Hو�ن�������-�C�D��'()*وא�KL�7אM�٠�

:قاعدة ���������

ث!+ + ن }س{دأ؛ ١؛ = نن؛!؛= ءس } س{د ن }س{دأ ت �������������������������

��@@�ýßbØnÛa@‡uëcpîÛbnÛa@òZ :مثال�����

ءس} ١+ @ س٣{ ) }٥ –س + # س{ت ~ ١ ����

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ـــ

ءس : ٣::+س:ذ:-:@:س [ } ١-س{ت ~۲

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ءس}٤+ذس+@س} {ذ–س {ت ~ ٣ ..........................................................................................................................................................................................................................................................................

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:١:+:#س[ @س ~٤ ........................................................................................................................................................................................................................................................................

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ءس@ } س٣ -# س۲ { :٤:+:#:س: :ذ :-:$س[# ت W��� @@ÝßbØnÛa@‡uëcتدريب��

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:قاعدة قاعدة قاعدة قاعدة �������

ث!++ن }ب+اس{؛ ١؛ = نن؛! ؛× أ؛= !ءس ن}ب+اس{ ت

Oא*و���ط��ن�6?�ن�א��א����+�א��(DQ ١-آل ن

��îÛbnÛa@pýßbØnÛa@‡uëc@@òZ :مثال

ء س ) )٧ + س ٣( ت ~١ �����

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ءس :٥:-:: س:ذ [ ت ~۲

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: أوجد التكامل: تدريبتدريبتدريبتدريب

....................................................................................................................................................................................................... = ءس% }س۲ - ٣{ت

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ءس @( )٤+ س ٤ - @س( ت ~٣

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ءس % )٧ + سس؛؛!؛ ( %س ت ~٤

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املتطابقات األساسية للتذكري

@ب+ ذ ا ب + @ ا= }@ ب +ا { : مربع جمموع حدين ~ ١

@ب+ ب ذ ا-@ ا= }@ ب -ا { : مربع الفرق بني حدين ~ ۲

@ ب-@ ا= } ب -ا} { ب +ا { : جمموع حدين يف الفرق بينهما ~ ٣

ب+ @ ا ب٣+ ب @ ا ٣+ # ا= }# ب + ا { : مكعب جمموع حدين ~ ٤

# ب-@ ا ب٣+ ب @ ا ٣ -# ا= }# ب -ا{ : مكعب الفرق بني حدين ~ ٥ ٧١

��א���� א� � �א��� א���م����

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:ل حاصل قسمة دالتني تكام: ثالثا

:حتليل البسط أو املقام واالختصار أل١

......... وميكن أن نستخدم حتليل فرق بني مربعني أو مكعبني أو جمموع مكعبني أو مكعب كامل أو مربع كامل ........ وذلك باستخدام املتطابقات

�òîÛbnÛa@pýßbØnÛa@‡uëc��W :مثال س ^؛؛؛؛؛؛؛؛؛؛؛؛٣+ ؛؛-سس؛؛؛ س%؛؛؛؛؛_؛؛@ س؛ ت ~١ ؛؛ ء

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ء س؛؛ ) ؛؛؛٦+؛ ؛= ؛ ؛ سس؛س^؛؛٢؛ ؛؛+؛ @ت س؛؛؛؛؛ ~٣

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.......................................................................................................................................................................................................................................................................... ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ

��Wأوجد التكامالت التالية: : : : تدريب تدريب تدريب تدريب

ء س ت ~۲ ء س ـــــــــــــــــــــــــــــــــــــــــت ~ ١

؛؛؛؛؛؛؛؛:::::::::+: :!؛٢ ؛؛؛؛؛؛؛سس؛# [ س ت ~٣ ؛؛؛ %؛؛ +؛ @ت س ~٤ ء س :::::؛؛؛؛؛سس ؛ ءس !؛ ؛$؛٢؛ _-؛س؛ سس

%)١-س( @)١-س(

(!} س -@ س { (! س

٧۲

��א���� א� � א���� א���م����

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: رفع املقام إىل البسط بإشارة سالبة أل۲

::::لتالية لتالية لتالية لتالية ويف هذه احلالة يكون عندنا احلاالت اويف هذه احلالة يكون عندنا احلاالت اويف هذه احلالة يكون عندنا احلاالت اويف هذه احلالة يكون عندنا احلاالت ا

�9�א��I(Rوא��?�/~ ا /��9�א��I(Rوא��?/��9�א��I(Rوא��?/�~ ب �9�א��I(Rوא��?-�C�D��E��دא��F�GHو��-�C�D��E��دא��F�GHو��-�C�D��E��دא��F�GHو��-�C�D��E��دא��F�GHج���������������������������������و� ~���������Oא*و���دא����+�א���/�?6�F�GHو������������Oא*و���دא����+�א���/�?6�F�GHو������������Oא*و���دא����+�א���/�?6�F�GHو������������Oא*و���دא����+�א���/�?6�F�GHو�������

�������������ýßbØnÛa@‡uëcpîÛbnÛa@òW�W�W�W : : : :مثالمثالمثالمثال

ء س � ت ~١

ث+ /)/س/( د[ ۲= ء س ت قاعدة ........................................................................................................................

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ــــــــــــــــــــــــــــــــــــــــــــــــــــس����������������������� ت~ ۲ ء

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٣+ س ۲ :٥::+س :٣:+:@س[

س{|د|| |{ /}س/{د [

٧ / /س/ /+/ خح٥[

٧٣

��א���� א� � א���� א���م����

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ء س ����������������� ت ~ ٣

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ـــــــــــــــــــــــــــــــــــــــــــــــــــــ

}س{د = ءس}س{دت ؛ء ؛سس �:نظرية

....................................... =} ٣{دæhÏ ث +/ص[= @؛ صص؛+ شش؛ ت س ¢sî} س{د=ص oãb×@aˆg :مثال

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ـــــــــــــــــــــــــــــــــــــــــــــــــــــ����:أوجد التكامل ::::تدريب تدريب تدريب تدريب

.........................................................................................................................................................................................= ء س ������������������������ ت

٥ - س #س

٧ ۲٥+ س١٠-@س

٧٤

��א���� א� � א���� א���م����

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òí‹öa‡Ûa@Þaë‡Ûa@ÝßbØm@@

<Ý^ŠÎ_<ì‚Â<±c<íén×n¹]<Ù]æ‚Ö]<løÚ^Ói<ÜéŠÏje<ÝçÏ߉<<

:تكامل الدوال املثلثية بصورة عامة

ث+ س جا = س ءس جتا ت ~ ۲ ث+ س جتا -= س ءس جا ت ~ ١

ث+ س قـا = س ظـا س قـا ت ~ ٤ ث+ س ظـا = س ءس @ قـا ت ~ ٣

ث+ سقتا -= س ءس ظتا س قتا ت ~٦ ث+ س ظتا -= س ءس @قتا ت ~ ٥ �òîÛbnÛa@pýßbØnÛa@‡uëcZ@: مثال

ءس} س جا+ سجتا{ ت ~١ ����........................................................................................................................................................................................................................................................................

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ـــــــــــــــــــــــــــــــــــــــــــــــ ءس}^ س @ جتا+ س @جا{ ت ~۲

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ــــــــــــــــــــــــــــــــــــــــــــــــــــــــ

ءس}سجتا+ @ س٣{ت ~٣

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٧٥

��א���� א� � א���� א���م����

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سءس ظاسقات ~ ٤

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ءس ]س@س جتا@قتا + س@س جا@قتا[ت ~ ٥

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بعض العالقات املهمة بعض العالقات املهمة بعض العالقات املهمة بعض العالقات املهمة

سجناس جا۲=س۲ جا ~۲ ١=س@جتا+س@جا ~١

س@قتا=١+س@ظتا ~٤ س @قا=١+س@ظا ~٣

ـــ = قاس ~٦ ـــــــــــــــــــــــ ـــــــــــــــــــــــ ـــــــــــــــــــــــ ـــــــــــــــــــــــ =قتاس ~٥

١ جاس جاس

١

٧٦

��א�������� א� � א���� א���م

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اليت زاويتها من الدرجة األوىل تكامل الدوال املثلثية أل١

ث+ س )ب + سا( جتا أ؛ !-= س ءس ) ب + س ا (جا ت ~ ١

ث+ س )ب + سا( جا أ؛= !س ءس ) ب + س ا (جتا ت ~ ۲

ث+ س )ب + سا(ظـا أ؛= !س ءس ) ب + س ا (@قـا ت ~ ٣

ث+ س )ب + س ا(قـا أ؛= !س ) ب + سا(ظـاس ) ب + س ا(قـا ت ~ ٤

ث+ س )ب + سا (ظتا أ؛ !-= س ءس ) ب + سا(@ قتا ت ~ ٥

ث+ س) ب + سا(قتا أ؛ !-= س ءس )ب + س ا ( ظتاس ) ب + س ا ( قتا ت ~٦

ث+ تكامل الدالة املثلثية بنفس الزاوية × ــــ =ل الدالة املثلثية اليت زاويتها من الدرجة األوىل تكام::::قاعدة قاعدة قاعدة قاعدة @@@@

@@@@@@@@@@@@@@@@@@@@òîmŁa@pýßbØnÛa@‡uëc@Z@

............................................................................................................................................................................................ =ءس)س ۲ –١(جتـات ~ ١

................................................................................................................................................................................................ =ءس)٥+ س ۲(جـا ت~ ۲

.......................................................................................................................................................................................... =ءس) س ۲ – ١( @قـا ت~ ٣

........................................................................................................................................................................................ =ءس ) ٣+ س ٤ (@قتـا ت ~ ٤

...........................................................................................................................................= ءس )١+ س ٣( ظـا ) ١+ س ٣( ـا ق ت~ ٥

..................................................................................................................................................= ءس )س ۲ –١( ظتـا )س۲– ١( قتـا ت ~٦

١ معامل س

٧٧

٧٧

��א���� א� � א���� א���م����

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ءس }سس ؛ ؛ ؛ الذ؛ %+سس ؛ ؛ةذ ؛؛!{ ت ~ ٧

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ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــت ~ ٨ ءس ـــــــــــــــ

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: : : :تدريبتدريبتدريبتدريب

ïÛbnÛa@ÝßbØnÛa@‡uëc���������W����������W����������W����������W� ءس س٥جا× س ٥ظات ����

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س٧ جا س٧@ جتا

٧٨

��א���� א� � א���� א���م����

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مشتقة الزاوية× تكامل الدوال املثلثية أل۲

ءس �א��T�T�א��א�� ت = ءس ) א��وא���א�D� ( × )��T�T�C��א�Sאو��( ت ��������: : : : قاعدا قاعدا قاعدا قاعدا

�òîÛbnÛa@pýßbØnÛa@‡uëcZ@: مثال

ءس @س٥س جا١٠ت ~١ �........................................................................................................................................................................................................................................................................

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ــــــــــــــــــــــــــــــــــــــــــــــــــــــــ ءسس٣ جتا٣ت ~ ۲

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ــــــــــــــــــــــــــــــــــــــــــــــــــــــــ ءس ) ٣+@س٥( ظتا ) ٣+@س٥( قتا س۲ت ~ ٣

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ــــــــــــــــــــــــــــــــــــــــــــــــــــــــ ��Wأوجد التكامل: تدريب ٧٩ ..................................................................................................................................................................................... = ءس#س @قا @ست

٧٧

��א���� א� � א���� א���م����

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:مشتقة الزاوية × تكامل الدوال املثلثية اس ن أل٣

ث + !+ن) دالة مثلثية( ١؛ = نن؛! ؛= ءسمشتقة الزاوية × مشتقتها × ن) دالة مثلثية ( ت

: : : : مالحظة مالحظة مالحظة مالحظة

بينما إذا كانت الدالتني ، س هو الدالة املراد تكاملها فصاحب األ) جتا ، جا ( إذا كانت الدالتني املضروبتني مها ٠فإن ظتا هي الدالة ) قتا ، ظتا ( فإن ظا هي الدالة وكذلك ) @قا، ظا ( املضروبتني مها

: : : : فمثال

١- آل ن ث + } س{ د !+ن جا؛ ١؛ = نن؛!؛= ءس } س{د } س{دجتا} س{ دن جات ~ �١

١- آل ن ث + } س{ د !+ن جتا؛ ١؛ = نن؛!؛= ءس } س{د } س{دجا} س{ دن جتات ~ �۲

�òîÛbnÛa@pýßbØnÛa@‡uëcW: مثال

س ءس٤ جتا س٤%جات ~ ١

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ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــ : أوجد التكامل��W تدريب

٨٠ ....................................................................................................................................................................................................= س ءس جا س &جتات

.

��א���� א� � א���� א���م����

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سءس@قا}$ ٧ –س ظا {ت ~ ۲

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ءس} ٥+@س٣@{قا} ٥+@س٣^{اظ س ت ~ ٣

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٧٧

��א���� א� � א���� א���م����

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:تكامل الدوال املثلثية من النوع أل٣

: لصورة عدد فردي موجب يف هذه احلالة نكتب التكامل على ان وكانت سنجـتا ، سنجـا

}س!_ ن جـتا × س جـتا {، } س!_ ن جـا × س جـا {

.١= س @جـتا + س @جـا: مث نستخدم املتطابقة

�òîÛbnÛa@pýßbØnÛa@‡uëcW: مثال

سء س #جـات ~١

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سء س #جـتات ~۲

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......................................................................................................................................................................................................................................................................... ٨۲

��א���� א� � א���� א���م����

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: ، ونستخدم التحويالت التالية س@ ظتا، س @ ظا، س @ جتا، س @ جا :تكامل الدوال املثلثية من النوع أل٤

س س@ جا ت ~١ ء س } س ۲ جتا – ١ { ٢؛ !ت= ء

ء س } س ۲ جتا+ ١ { ٢؛ !ت= ء س س@ جتا ت ~۲

ء س } ١ -س @ قـا { ت= ء س س@ ظا ت ~٣

ء س } ١ -س @ قتا { ت= ء س س@ ظتا ت ~٤

: مالحظات

: يف القوانني فوق فهي من األخطاء اليت يقع فيها الطالب مثل äfjÞ]ØèçvjÖ]<ð^ßm_<íèæ]ˆ×Ö>~ ا .}س ٨ جتا – ١ { ٢؛! = س٤@ جا

...... ،٦ ، ٤ ، ۲: نستخدم نفس القوانني إذا كان األس زوجي أي ~ ب

:òîÛbnÛa@pýßbØnÛa@‡uëc: مثال

سء س @جـات ~١

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: أوجد التكامل�����W تدريب

.....................................................................................................................................................................................................= سء س ٢؛! @ـاتجت ٨٣

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س } س٣@{ظات ~ ذ ء

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ءس }س@جتا+س@جا+س@ظا{ت~ ٣

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ء س)٣-س۲( @ظتا ت ~٤

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: أوجد التكامل�����W تدريب

٨٤ .....................................................................................................................................................................................= سء س ٢؛! @جتـات

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التكامل بالتعويض

::::ابقتني ابقتني ابقتني ابقتني وهي للمسائل اليت ليست على الصورتني السوهي للمسائل اليت ليست على الصورتني السوهي للمسائل اليت ليست على الصورتني السوهي للمسائل اليت ليست على الصورتني الس: : : : طريقة التعويض طريقة التعويض طريقة التعويض طريقة التعويض أل٥

ونفكر يف طريقة التعويض إذا كان التكامل حاصل ضرب دالتني وال نستطيع أن نضرب الدالتني يف بعض وليست . على صورة دالة يف مشتقتها فنلجأ إىل طريقة التعويض

.ء س بـ ء ص واستبدال ص هو استبدال الدالة املعطاة بدالة جديدة يف املتغري

.ن تساوي القيمة اليت حتت اجلذر أو داخل القوس أس ص دائما نفرض : مالحظة

�ïÛbnÛa@ÝßbØnÛa@‡uëcW : مثال

ء س / ۲ /-/س [@ س ت ~١

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ءس /٣/ /+/ خحس[} ٥+ س {ت ~ ۲

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ء س*}ذ + س { ذس ت ~۲ ء س %}١ + ؛ سس؛!^ { س ت ~١ �Wو���א��?�/�א���=�:تدريب ٨٦

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†‡�a@�Ë@ÝßbØnÛa@óÜÇ@pbÔîjĐm@ :تطبيقات هندسية : أوال

والفكرة كلها تدور حول إجياد قيمة، يف التطبيقات اهلندسية سندرس طريقة إجياد معادلة املنحىن اليت حتقق شرطا ما ٠ الثابت ث

:خطوات إجياد معادلة املنحىن أعطي ونقطة عليه

٠ نكامل املشتقة لنحصل على الدالة ويظهر فيها الثابت ث ~ا

٠نعوض بالنقطة املعطاة يف السؤال يف الدالة اليت وجدا يف اخلطوة األوىل ونوجد قيمة ث ~ ب

٠صلية إذا وجدنا قيمة ث نعوض بقيمتها يف الدالة بعد التكامل فنحصل على الدالة األ~ ج

:مالحظة

٠ميل املماس يعين املشتقة األوىل أل١

ب = } ا{دفإن } ا،ب {إذا مر املنحىن بالنقطة أل۲

ب = } ا{دفإن } ا،ب {إذا مر املنحىن بالنقطة أل٣

ءس} س{د ت = } س{د ، ءس } س{د ت = } س{د ت أل٤

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�.VC��@ أل١ Q�(W�XKא��Y.Z.د���א����و��/�`_^�F١�،٣�E��VC8�/0��.Gאو����� س١٠ -= د <�������������������������

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وX }س،ص{ VC8�X���.G�} س{د= ص 9ذא�0ن���/�א�.Y.Z أل٣ e��^��G���Hس ، ۲ @@وא�Y.Z.وא�������������������(W��VC.�Q }-١ ،۲ { ،=>�Y.Z.د���א�@........+ ....... =ص fcن���

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٧ ~ د ٣~ ج١~ ب صفر ~ا

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:تطبيقات فيزيائية: ثانيا

٠ يف التطبيقات الفيزيائية سندرس طريقة إجياد املسافة إذا أعطيت السرعة أو التسارع

ن، ت دالة يف الزمن ع كل من ف ،ع ء ن ت = ف ء ن ، ت ت = ع : معادلة املسافة خطوات إجياد

٠ إذا أعطي التسارع نكامل فنحصل على السرعة~ا

٠ نوجد قيمة ث مبعلومية السرعة والزمن من بدء احلركة ~ ب

٠ نكامل السرعة فنحصل على املسافة وإلجياد قيمة الثابت ث نعوض بالزمن عند بدء احلركة ~ ج

شتقاق ا ئ اشتقاق ئ اشتقاق ئ }ن{التسارع ت } ن{ع السرعة }ن {اإلزاحة ف تكامل ؤ تكامل ؤ تكامل ؤ

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. ٤= ن إىل ذ = ن �و���9زא���א���eL(�ل�א�@��Fi+ ذ ن + @ ن٣= ع

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ة ثوان من بدء احلرك٨ث ، فأوجد إزاحة اجلسيم بعد / م١٠ ثوان تساوي ٥ بعد ماجلسي ٩۲

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عند ٤+ذن -@ ن٣= } ن{ع جلسيم يتحرك بسرعة } ن{أوجد دالة املسافة ف ::::تدريب تدريب تدريب تدريب ٩٤ ٠ ٦= } ٠{ علمت أن ف إذا ن ، الزمن

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†‡�a@ÝßbØnÛa@@

] : ا ، ب [جتزئية الفترة املغلقة: أوال

النقاط تسمى من الفترات اجلزيئية ويتم ذلك بتحديد جمموعة منن أن جتزئ هذه الفترة يعين تقسيمها إىل عدد قدره

٠) التجزئ (نقاط التقسيم

٠ س =ا ١ س ٢ س ٣س س ننس = ب

} ننس ، ... ، س ، ... ، ٣ ، س ٢ ، س ١ ، س ٠س { : نقط التجزيء

:ونكتب } ا ، ب { نن ونرمز له بالرمز ت } ] ا ، ب [التجزيء النوين للفترة { يسمى التجزيء الناتج

)اية الفترة( ب = ننسو )بداية الفترة( ا = ٠س حيث } ننس، ... ، ٣ ، س ٢ ، س ١ ، س ٠س { =}ا ، ب { ننت :ينقسم جتزئ الفترات اىل جزئني

جتزئ غري منتظم أل١ ٠لتجزئ يكون أطوال الفترات غري متساو ويف هذا ا

جتزئ منتظم أل۲

٠ ويف هذا التجزئ يكون أطوال الفترات اجلزئية مجيعها متساو

موستكون اغلب دراستنا على التجزئي النوين املنتظ

٩٥

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:::: مالحظات

١+ ن = أي عدد نقاط التجزئ ١+ عدد الفترات = عدد نقاط التجزئ ~ ١

٠كل عنصر من نقاط التجزئ أصغر من العنصر الذي قبله ~ ۲

٠كذا وه....... ٢ س : والثالث هو ١ س :والثاين هو ٠س :العنصر األول من عناصر التجزئ هو ~ ٣

٠وهكذا ................. ]٢ ، س ١س [: والثانية ، ]١ ، س ٠س [: الفترة األوىل ~٤

فمثال: للداللة على طول الفترة مم س: يستخدم الرمز ~٥

١مم س + ٠س = ١س ئ ٠ س – ١س = ١مم س : طول الفترة األوىل

٢مم س + ١س = ٢س ئ ١ س – ٢س = ٢ مم س :طول الفترة الثانية

١ - س – س = س مم= تسمى الفترة الرائية وطوهلا ]١ - ، س س [الفترة ~ ٦

} ] ا ، ب [ التجزيء املنتظم النوين للفترة{ متساوية سمي التجزيء الناتج إذا كانت املسافات بني نقط التجزيء ~٧

}س مم× ر + ا = س : الحظ أن {؛ نن؛ا؛ _ ب؛ ؛= س مم: وسيكون حينها طول الفترة اجلزيئية

: : : : حيث حيث حيث حيث

٥ = ر ئسادس العنصر ال: فمثال ... العنصر املطلوب ويقل عن الرتبة بواحد : ر

٠اية الفترة : ب ،بداية الفترة :ا، طول الفترة : مم س

٩٦

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٧٧

��א� א� � א���� א���م�������

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٧ ~oãb×@aˆg ٠ مجس} س{ د ÇóÜ] ب ٠ مجسء س } س{د ب تأ : @æhÏ] ا ،

٨ ~oãb×@aˆg ب آا æhÏ : ‘ء س‘}س{د‘ ب تأ محس ‘ء س } س{د ب تأ

٩ ~æb×@aˆg ٢ ، د ١ د كال من ñ�ÐÛa@óÜÇ@ÝßbØnÜÛ@òÜibÓ] ٢ د مجس ١ د وكان ] ا ، ب æhÏ: ء س} س{٢د ب تأ مجس س ء} س{١د ب تأ

مثالمثالمثالمثال Fi@א��Y�G�/�?������QH�78א��א����د� ٧ مجس} س{د و078 ، ٥ ، ۲ أ 9ذא�0وX@@@ءس}س{د ٣% ت ٢ fcن����C,����,�H�(oא ٥ ، ۲ أ يس �?/ e6���� ٦٣ ~ د ۲٤~ ج ۲١~ ب٩ ~ا

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::::واجب واجب واجب واجب

.....= ءس }س{د٥# ت ٥ ، فإن ٨ = ءس}س{د% ت ٣ إذا كان

١١۲ ٤٠ ~ د ٥~ ج٥ - ~ ب ٤٠ - ~ا

��א���� א� � א���� א���م����

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....................................................................................... @uëdχ ،٧ = سء)س(ه# ت١ ، ٥ = سء)س(د# ت١ : æb×@aˆg مثالمثالمثالمثال

سء ) س(ه ۲) + س(د٣أ !ت٣ ~ج سء ) س(ه ٤ -)س(د٣أ# ت١ ~ب سء )س(ه)+ س(دأ# ت١ ~ا

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س }س{ه @ت آإذا كان :::: تتتتددددررررييييبببب = ك فإن قيمة ١٠= ء س }٣+ } س { ه { ى ت ٢، ٥ =ء ب ٤~ ا ٧ ~ د ٦<~ ٥ ~

١١٣

٧٧

��א���� א� � א���� א���م����

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.....= ءس سس ؛ !لو % ت ١+ س ءس لو % ت ١ مثالمثالمثالمثال

٤ لو~ د٥لو-١لو~ ج ٠~ ب ١لو -٥لو ~ ا

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ن مثالمثالمثالمثال �VC.�78א�,�ن��Y.Z.��O9א��א��، } ١-، ٣{ ، } ٨-، ١{ 9ذא�0�6. س }س{د # ت ١ fcن} س{د=ص ٩- ~ د ٧ -~ ج ٧~ ب ٩ ~ا .ء

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ـــــــــــــــــــــــــــــــــــــــــــــــــــــ : تدريبتدريبتدريبتدريب ءس }س{د # ت ١ فإن ٩=} ٣{د، ٥= }١{د وكان، ٣ ،١ أ متصلة على الفترةد إذا كانت الدالة

١١٤ ٤ ~ د ٣٥~ ج ۲- ~ ب ۲ ~ا

��א���� א� � א���� א���م����

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ÝßbØnÜÛ@òĐ�ìn¾a@òàîÔÛa@òí‹Äã@

: حبيث أا،ب ي ٠س فإنه يوجد نقطة أا،ب إذا كانت الدالة د متصلة يف

} ٠س {د} ا–ب {= ءس }س{دت يب ]

:التفسري اهلندسي

وحمور السيناتب= ا ، س= سواحملصورة باملستقيمني } س{دعة حتت املنحىن هي املساحة الواق ءس }س{دت يب ]

} ٠س {، د} ا–ب {هي مساحة املستطيل الذي بعداه } ٠س {د} ا–ب { ـــــــــــــــــــــــــــــــــــــــــــــــــــــــ

:مثال حملول

ء س}٣ -ذس {# ٠ تينه نظرية القيمة املتوسطة للتكامل الذي تع٠س أوجد قيمة العدد

] ٣ ، ٠[ ، ٣ -ذس = } س{د :احلل

الدالة متصلة وقابلة للتكامل على الفترة املعطاة ألا كثرية حدود

صفر = } ٠-٠ { –} ٩-٩{ = # ٠ س ٣ -@ أ س= ء س } ٣ –ذس {# ٠ ت~ ١

٣ – ٠ذس = }٠س { د~ ذ

}٠س { د} ٠ – ٣{ = ء س } ٣ –ذس {# ٠تإ

٢؛# = ٠ ئ س ٦؛ = )٠ ئ س ٩ = ٠س ٦ئ ٩ – ٠س ٦= ٠ئ } ٣ – ٠ذس } { ٠ – ٣{ = ٠ ئ

/ / / / مالحظة مالحظة مالحظة مالحظة

��8)���א��C,��א���_V���9���٠ذא�0�78א��א����+�א�����א*وfc�Oن��H,��س�-CCM�pא��/�?������������E6?�ن����Fi@א��} �.��٠�١١٥

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��א���� א� � א���� א���م����

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���א������?�/�@@���6�XK.^��8)���א��C,��א���_V�<��א���د�אذ = ٠ س 9ذא�078 أل١ @. ل bcو����H,�، ] ٣ ، ٠[ א�@Y�G������������Fi ٦+ س ٤ -@ سل = } س{د

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��א���� א� � א���� א���م����

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....................................................................................... }٠س {د bcو����H,� ، ٦=} س{د@ ت ٠ و0ن ، ۲، ٠أ �� ����Y�Gא�@Fi }س{د 9ذא�078 � أل۲

#�x ٠ ۲، ٠أ ي ٠س

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��א���� א� � א���� א���م����

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�/^�א���6�XK.د<��א��� ٠س 9ذא�0ن أل٣ ?�����V_��א���,�Cس{د ���א���@@���8)���א� {@@Fi@א��Y�G�@...........= ءس}س{ د% ت ٢ :@fcن ٤= } ٠س {د و078 ٥، ۲أ ���

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هو العدد الذي يحقق نظرية القيمة المتوسطة في التكامل للدالة ٠س إذا كان تدريبتدريبتدريبتدريب

: =} ٤{د فإنذ ، +٠ س ٣= ء س }س{د @ ٠ت بحيث ذ،٠أ على الفترة} س{د ١١٨ ١٤~ د ٧~ ج ٤~ ب ذ ~ ا

��א���� א� � א���� א���م����

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ÝßbØnÜÛ@lb�¨@òî�b�þa@òí‹ÄäÛa@

:نظرية

: وكانت ي هي الدالة املعرفة بالعالقة أا،ب متصلة يف } س{د إذا كانت الدالة

}س{د = }س{ فإن ي أا،ب يس لكل ءع }ع {د مسست ]= } س{ يـــــــــــــــــــــــــــــــــــــــــــــــــــ ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ ـــ

:مثال حملول

}س{ى ء؛؛؛ ؛ ؛ سس؛ فأوجدء ع } ٥+ ع ٣+ @ ع { ست ١ = } س { إذا كانت ى : احلل

اجلزء األول–من النظرية األساسية للتفاضل والتكامل

٥+ س ٣+ @ س= }س { د = } س { ى ؛ ء؛؛؛ ؛ ؛ سس

ـــــــــــــــــــــــــــــــــــــــــــــــــــ ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ ــــــــ :مثال حملول

.}س { د فأوجد ٦+ @ س٦= ء ع } ع { د ست ١ إذا كان

اجلزء األول–من النظرية األساسية للتفاضل والتكامل :احلل

س١۲= } ٦+ @ س٦{ ؛ ء؛؛؛ ؛ ؛ سس = ع ء} ع { د ست ١ ؛ ء؛؛؛ ؛ ؛ سس = }س { د ــــــــــــــــــــــــــــــــــــــــــــــــــــــــ

)٥(فاوجد د ٦ -@ س٦= ء ع )ع( د مسس ت ١إذا كان مثال مثال مثال مثال

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��א���� א� � א���� א���م����

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78 ......= } }۲ه fcنء ع { ١- ع۲+@ ع { مسس ت ١= } س{ه 9ذא�0

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UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU���� : أكمل ����������������

.....= } س{هإن فء ع )ع( د مسس ت ١= } س{ه إذا كانت ~ا

= .........}١{ه فإن ء ع ؛؛؛؛؛؛؛؛؛؛؛ ١؛؛؛؛؛؛؛؛؛؛؛؛؛؛؛؛؛=ع ؛ عع ؛؛؛؛؛؛ مسس ت ٠= } س{ه إذا كانت ~ب

.......) =س(فإن د ظا س = ء ع )ع( د مسس ت ١= } س{هإذا كانت ~ج ــــــــــــــــــــــــــــــــــــــــــــــــــــ

:= } ۲{ق فإن ءع } ٥ –ذع + @ ع { مسست ١ = } س{ق oãb×@aˆg@@ تدريبتدريبتدريبتدريب ٦~ د ٣~ ج صفر ~ ب ٥-~ ا

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