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  إجابات كتاب التمارين

إجابات كتاب التمارين

تكامل اقترانات خاصة

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral fraction numerator 1 minus x squared over denominator 5 x end fraction d x end style (1)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral fraction numerator 1 minus x squared over denominator 5 x end fraction d x end cell cell equals integral left parenthesis fraction numerator 1 over denominator 5 x end fraction minus fraction numerator x squared over denominator 5 x end fraction right parenthesis d x end cell row blank cell equals integral left parenthesis fraction numerator 1 over denominator 5 x end fraction minus 1 fifth x right parenthesis d x end cell row blank cell equals 1 fifth ln invisible function application vertical line x vertical line minus 1 over 10 x squared plus C end cell end table end style

begin mathsize 20px style integral left parenthesis 5 e to the power of x plus 4 right parenthesis d x end style (2)

begin mathsize 20px style integral left parenthesis 5 e to the power of x plus 4 right parenthesis d x equals 5 e to the power of x plus 4 x plus C end style

begin mathsize 20px style integral left parenthesis 1 minus e to the power of 2 x minus 3 end exponent right parenthesis d x end style (3)

begin mathsize 20px style integral left parenthesis 1 minus e to the power of 2 x minus 3 end exponent right parenthesis d x equals x minus 1 half e to the power of 2 x minus 3 end exponent plus C end style

begin mathsize 20px style integral left parenthesis sin invisible function application 2 x minus cos invisible function application 2 x right parenthesis d x end style (4)

begin mathsize 20px style integral left parenthesis sin invisible function application 2 x minus cos invisible function application 2 x right parenthesis d x equals negative 1 half cos invisible function application 2 x minus 1 half sin invisible function application 2 x plus C end style

begin mathsize 20px style integral fraction numerator 3 over denominator 2 x minus 1 end fraction d x end style (5)

begin mathsize 20px style integral fraction numerator 3 over denominator 2 x minus 1 end fraction d x equals 3 over 2 ln invisible function application vertical line 2 x minus 1 vertical line plus C end style

begin mathsize 20px style integral left parenthesis 5 minus sin invisible function application left parenthesis 5 minus 5 x right parenthesis right parenthesis d x end style (6)

begin mathsize 20px style integral left parenthesis 5 minus sin invisible function application left parenthesis 5 minus 5 x right parenthesis right parenthesis d x equals 5 x minus 1 fifth cos invisible function application left parenthesis 5 minus 5 x right parenthesis plus C end style

begin mathsize 20px style integral fraction numerator 1 over denominator 1 third x minus 2 end fraction d x end style (7)

begin mathsize 20px style integral fraction numerator 1 over denominator 1 third x minus 2 end fraction d x equals 3 ln invisible function application vertical line 1 third x minus 2 vertical line plus C end style

begin mathsize 20px style integral left parenthesis 2 x minus 1 plus fraction numerator 8 over denominator 5 x plus 4 end fraction right parenthesis d x end style (8)

begin mathsize 20px style integral left parenthesis 2 x minus 1 plus fraction numerator 8 over denominator 5 x plus 4 end fraction right parenthesis d x equals x squared minus x plus 8 over 5 ln invisible function application vertical line 5 x plus 4 vertical line plus C end style

begin mathsize 20px style integral left parenthesis 3 cos invisible function application x plus 5 over x plus 4 over x squared right parenthesis d x end style (9)

begin mathsize 20px style integral left parenthesis 3 cos invisible function application x plus 5 over x plus 4 over x squared right parenthesis d x equals 3 sin invisible function application x plus 5 ln invisible function application vertical line x vertical line minus 4 over x plus C end style

begin mathsize 20px style integral left parenthesis 3 x plus 2 right parenthesis to the power of 5 d x end style (10)

begin mathsize 20px style integral left parenthesis 3 x plus 2 right parenthesis to the power of 5 d x equals 1 over 18 left parenthesis 3 x plus 2 right parenthesis to the power of 6 plus C end style

begin mathsize 20px style integral fraction numerator x plus 1 over denominator x squared plus 2 x plus 5 end fraction d x end style (11)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral fraction numerator x plus 1 over denominator x squared plus 2 x plus 5 end fraction d x end cell cell equals 1 half integral fraction numerator 2 x plus 2 over denominator x squared plus 2 x plus 5 end fraction d x end cell row blank cell equals 1 half ln invisible function application vertical line x squared plus 2 x plus 5 vertical line plus C end cell end table end style

begin mathsize 20px style integral left parenthesis e to the power of 2 x end exponent minus 1 half sin invisible function application left parenthesis 2 x minus 1 right parenthesis right parenthesis d x end style (12)

begin mathsize 20px style integral left parenthesis e to the power of 2 x end exponent minus 1 half sin invisible function application left parenthesis 2 x minus 1 right parenthesis right parenthesis d x equals 1 half e to the power of 2 x end exponent plus 1 fourth cos invisible function application left parenthesis 2 x minus 1 right parenthesis plus C end style

begin mathsize 20px style integral left parenthesis sin invisible function application left parenthesis 2 x plus 3 right parenthesis plus cos invisible function application left parenthesis 3 x plus 2 right parenthesis right parenthesis d x end style (13)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral left parenthesis sin invisible function application left parenthesis 2 x plus 3 right parenthesis plus cos invisible function application left parenthesis 3 x plus 2 right parenthesis right parenthesis d x end cell row blank cell equals negative 1 half cos invisible function application left parenthesis 2 x plus 3 right parenthesis plus 1 third sin invisible function application left parenthesis 3 x plus 2 right parenthesis plus C end cell end table end style

begin mathsize 20px style integral left parenthesis 1 over 8 x to the power of 3 divided by 2 end exponent minus 4 over x right parenthesis d x end style (14)

begin mathsize 20px style integral left parenthesis 1 over 8 x to the power of 3 over 2 end exponent minus 4 over x right parenthesis d x equals 1 over 20 x to the power of 5 over 2 end exponent minus 4 ln invisible function application vertical line x vertical line plus C end style

begin mathsize 20px style integral fraction numerator 1 over denominator square root of x minus 1 end root end fraction d x end style (15)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral fraction numerator 1 over denominator square root of x minus 1 end root end fraction d x end cell cell equals integral left parenthesis x minus 1 right parenthesis to the power of negative 1 half end exponent d x end cell row blank cell equals 2 left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent plus C end cell row blank cell equals 2 square root of x minus 1 end root plus C end cell end table end style

أجد قيمة كل من التكاملات الآتية:

begin mathsize 20px style integral subscript 0 superscript 1 square root of 1 plus 7 x end root d x end style (16)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral subscript 0 superscript 1 square root of 1 plus 7 x end root d x end cell cell equals integral subscript 0 superscript 1 left parenthesis 1 plus 7 x right parenthesis to the power of 1 half end exponent d x end cell row blank cell equals 2 over 21 left parenthesis 1 plus 7 x right parenthesis to the power of 3 over 2 end exponent vertical line subscript 0 superscript 1 end cell row blank cell equals 2 over 21 left parenthesis 1 plus 7 left parenthesis 1 right parenthesis right parenthesis to the power of 3 over 2 end exponent minus 2 over 21 left parenthesis 1 plus 7 left parenthesis 0 right parenthesis right parenthesis to the power of 3 over 2 end exponent end cell row blank cell equals 2 over 21 square root of 512 minus 2 over 21 end cell end table end style

begin mathsize 20px style integral subscript 0 superscript 1 e to the power of x left parenthesis 4 minus e to the power of x right parenthesis d x end style (17)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral subscript 0 superscript 1 e to the power of x left parenthesis 4 minus e to the power of x right parenthesis d x end cell cell equals integral subscript 0 superscript 1 left parenthesis 4 e to the power of x minus e to the power of 2 x end exponent right parenthesis d x end cell row blank cell equals left parenthesis 4 e to the power of x minus 1 half e to the power of 2 x end exponent right parenthesis vertical line subscript 0 superscript 1 end cell row blank cell equals 4 e minus 1 half e squared minus 7 over 2 end cell end table end style

begin mathsize 20px style integral subscript 1 superscript 3 left parenthesis 1 plus 1 over x right parenthesis d x end style (18)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral subscript 1 superscript 3 left parenthesis 1 plus 1 over x right parenthesis d x end cell cell equals left parenthesis x plus ln invisible function application vertical line x vertical line right parenthesis vertical line subscript 1 superscript 3 end cell row blank cell equals 2 plus ln invisible function application 3 end cell end table end style

(19) إذا كان ميل المماس لمحنى العلاقة هوbegin mathsize 20px style y end style: begin mathsize 20px style b b b end style، فأجد قاعدة العلاقة begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 6 e to the power of 2 x end exponent plus 2 e to the power of negative x end exponent end style، علماً بأن منحناها يمر بالنقطة begin mathsize 20px style left parenthesis 0 comma 2 right parenthesis end style.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y equals f left parenthesis x right parenthesis equals integral left parenthesis 6 e to the power of 2 x end exponent plus 2 e to the power of negative x end exponent right parenthesis d x end cell row blank cell equals 3 e to the power of 2 x end exponent minus 2 e to the power of negative x end exponent plus C end cell row blank cell y equals f left parenthesis x right parenthesis equals 3 e to the power of 2 x end exponent minus 2 e to the power of negative x end exponent plus C end cell row blank cell f left parenthesis 0 right parenthesis equals 2 not stretchy rightwards double arrow 3 minus 2 plus C equals 2 end cell row blank cell not stretchy rightwards double arrow C equals 1 end cell row blank cell not stretchy rightwards double arrow y equals 3 e to the power of 2 x end exponent minus 2 e to the power of negative x end exponent plus 1 end cell end table end cell end table end style

في كل مما يأتي المشتقة الأولى للاقتران begin mathsize 20px style bold italic f bold left parenthesis bold italic x bold right parenthesis end style، ونقطة يمر بها منحنى begin mathsize 20px style bold italic y bold equals bold italic f bold left parenthesis bold italic x bold right parenthesis end style. أستعمل المعلومات المعطاة لإيجاد قاعدة الاقتران begin mathsize 20px style bold italic f bold left parenthesis bold italic x bold right parenthesis end style:

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals e to the power of negative x end exponent semicolon left parenthesis 0 comma 3 right parenthesis end style (20)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral e to the power of negative x end exponent d x equals negative e to the power of negative x end exponent plus C end cell row blank cell f left parenthesis 0 right parenthesis equals 3 not stretchy rightwards double arrow negative 1 plus C equals 3 end cell row blank cell space of 1em not stretchy rightwards double arrow C equals 4 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative e to the power of negative x end exponent plus 4 end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 3 over x minus 4 semicolon left parenthesis 1 comma 0 right parenthesis end style (21)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell f left parenthesis x right parenthesis equals integral left parenthesis 3 over x minus 4 right parenthesis d x end cell row blank cell equals 3 ln invisible function application vertical line x vertical line minus 4 x plus C end cell row blank cell f left parenthesis 1 right parenthesis equals 0 not stretchy rightwards double arrow negative 4 plus C equals 0 end cell row blank cell space of 1em not stretchy rightwards double arrow C equals 4 end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 3 ln invisible function application vertical line x vertical line minus 4 x plus 4 end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals 4 e to the power of x minus 2 semicolon left parenthesis 0 comma 1 right parenthesis end style (22)

begin mathsize 20px style table attributes columnspacing 1em end attributes row cell f left parenthesis x right parenthesis equals integral left parenthesis 4 e to the power of x minus 2 right parenthesis d x end cell row cell equals 4 e to the power of x minus 2 x plus C end cell row cell f left parenthesis 0 right parenthesis equals 1 not stretchy rightwards double arrow 4 plus C equals 1 end cell row cell not stretchy rightwards double arrow C equals negative 3 end cell row cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 4 e to the power of x minus 2 x minus 3 end cell end table end style

(23) تلوث: يعالج التلوث في بحيرة باستعمال مضاد للبكتيريا. إذا كان عـدد الخلايا البكتيرية الضارة لكل مليلتر من الماء في البحيرة يتغير بمعدل: begin mathsize 20px style N to the power of straight prime left parenthesis t right parenthesis equals negative fraction numerator 2000 t over denominator 1 plus t squared end fraction end style، حيث begin mathsize 20px style N left parenthesis t right parenthesis end style عدد الخلايا البكتيرية لكل مليلتر من الماء بعد begin mathsize 20px style t end style يوماً من استعمال المضاد، فأجد begin mathsize 20px style N left parenthesis t right parenthesis end style، علماً بأن العدد الابتدائي للخلايا هو 5000 خلية لكل مليلتر.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell s left parenthesis t right parenthesis equals integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t end cell row blank cell u equals t squared plus 1 not stretchy rightwards double arrow fraction numerator d u over denominator d t end fraction equals 2 t not stretchy rightwards double arrow d t equals fraction numerator d u over denominator 2 t end fraction end cell row blank cell integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t equals integral u to the power of negative 1 half end exponent fraction numerator d u over denominator 2 t end fraction equals integral 1 half u to the power of negative 1 half end exponent d u equals u to the power of 1 half end exponent plus C equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 1 plus C equals 0 not stretchy rightwards double arrow C equals negative 1 end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root minus 1 end cell end table end style

(24) أحدد أوجه الاختلاف بين التكاملين الآتيين من دون إيجاد التكامل:

صورة للسؤال 24

التكامل الأيسر هو مجموع تكاملين لاقترانين، أحدهما مثلثي هو begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell s left parenthesis t right parenthesis equals integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t end cell row blank cell u equals t squared plus 1 not stretchy rightwards double arrow fraction numerator d u over denominator d t end fraction equals 2 t not stretchy rightwards double arrow d t equals fraction numerator d u over denominator 2 t end fraction end cell row blank cell integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t equals integral u to the power of negative 1 half end exponent fraction numerator d u over denominator 2 t end fraction equals integral 1 half u to the power of negative 1 half end exponent d u equals u to the power of 1 half end exponent plus C equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 1 plus C equals 0 not stretchy rightwards double arrow C equals negative 1 end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root minus 1 end cell end table end style والآخر ثابت هو begin mathsize 20px style g left parenthesis x right parenthesis equals 1 end style.

بينما التكامل الأيمن هو اتقتران مثلثي واحد فقط هو begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell s left parenthesis t right parenthesis equals integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t end cell row blank cell u equals t squared plus 1 not stretchy rightwards double arrow fraction numerator d u over denominator d t end fraction equals 2 t not stretchy rightwards double arrow d t equals fraction numerator d u over denominator 2 t end fraction end cell row blank cell integral fraction numerator t over denominator square root of t squared plus 1 end root end fraction d t equals integral u to the power of negative 1 half end exponent fraction numerator d u over denominator 2 t end fraction equals integral 1 half u to the power of negative 1 half end exponent d u equals u to the power of 1 half end exponent plus C equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root plus C end cell row blank cell s left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 1 plus C equals 0 not stretchy rightwards double arrow C equals negative 1 end cell row blank cell s left parenthesis t right parenthesis equals square root of t squared plus 1 end root minus 1 end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 02 / 2023

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