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إجابات كتاب التمارين

التكامل بالتعويض

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral x square root of x squared plus 3 end root 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 blank cell integral x square root of x squared plus 3 end root d x end cell row blank cell u equals x squared plus 3 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 2 x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 2 x end fraction end cell 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 cell integral x square root of x squared plus 3 end root d x equals integral x u to the power of 1 half end exponent fraction numerator d u over denominator 2 x end fraction end cell cell equals integral subscript 1 half end subscript superscript 1 half u to the power of 1 half end exponent end superscript d u end cell row blank cell equals 1 third u to the power of 3 over 2 end exponent plus C equals 1 third square root of left parenthesis x squared plus 3 right parenthesis cubed end root plus C end cell end table end cell end table end style

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

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 x to the power of 4 e to the power of x to the power of 5 plus 2 end exponent d x end cell 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 u equals x to the power of 5 plus 2 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 5 x to the power of 4 not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 5 x to the power of 4 end fraction end cell row blank cell integral x to the power of 4 e to the power of x to the power of 5 plus 2 end exponent d x equals integral x to the power of 4 e to the power of u fraction numerator d u over denominator 5 x to the power of 4 end fraction equals integral subscript 0 1 fifth e to the power of u d u end cell row blank cell equals 1 fifth e to the power of u plus C equals 1 fifth e to the power of x to the power of 5 plus 2 end exponent plus C end cell end table end cell end table end style

begin mathsize 20px style integral left parenthesis x plus 1 right parenthesis left parenthesis x squared plus 2 x plus 5 right parenthesis to the power of 4 d x end style (3)

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 x plus 1 right parenthesis left parenthesis x squared plus 2 x plus 5 right parenthesis to the power of 4 d x end cell row blank cell u equals x squared plus 2 x plus 5 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 2 x plus 2 not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 2 x plus 2 end fraction end cell row blank cell integral left parenthesis x plus 1 right parenthesis left parenthesis x squared plus 2 x plus 5 right parenthesis to the power of 4 d x equals integral left parenthesis x plus 1 right parenthesis u to the power of 4 fraction numerator d u over denominator 2 x plus 2 end fraction equals integral 1 half u to the power of 4 d u end cell row blank cell equals 1 over 10 u to the power of 5 plus C equals 1 over 10 left parenthesis x squared plus 2 x plus 5 right parenthesis to the power of 5 plus C end cell end table end style

begin mathsize 20px style integral fraction numerator left parenthesis ln invisible function application x right parenthesis cubed over denominator x end fraction d x end style (4)

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 fraction numerator left parenthesis ln invisible function application x right parenthesis cubed over denominator x end fraction d x end cell row blank cell u equals ln invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 1 over x not stretchy rightwards double arrow d x equals x d u end cell 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 cell integral fraction numerator left parenthesis ln invisible function application x right parenthesis cubed over denominator x end fraction d x equals integral u cubed over x x d u end cell cell equals integral subscript 0 u cubed d u end cell row blank cell equals 1 fourth u to the power of 4 plus C equals 1 fourth left parenthesis ln invisible function application x right parenthesis to the power of 4 plus C end cell end table end cell end table end style

begin mathsize 20px style integral fraction numerator cos invisible function application x over denominator sin to the power of 4 invisible function application x end fraction d x end style (5)

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 fraction numerator cos invisible function application x over denominator sin to the power of 4 invisible function application x end fraction d x end cell row blank cell u equals sin invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals cos invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator cos invisible function application x end fraction end cell row blank cell integral fraction numerator cos invisible function application x over denominator sin to the power of 4 invisible function application x end fraction d x equals integral fraction numerator cos invisible function application x over denominator u to the power of 4 end fraction fraction numerator d u over denominator cos invisible function application x end fraction equals integral u to the power of negative 4 end exponent d u end cell row blank cell equals negative 1 third u to the power of negative 3 end exponent plus C equals negative 1 third left parenthesis sin invisible function application x right parenthesis to the power of negative 3 end exponent plus C end cell end table end style

begin mathsize 20px style integral sin invisible function application x square root of 1 plus 3 cos invisible function application x end root d x end style (6)

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 sin invisible function application x square root of 1 plus 3 cos invisible function application x end root d x end cell row blank cell u equals 1 plus 3 cos invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals negative 3 sin invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator negative 3 sin invisible function application x end fraction end cell row blank cell integral sin invisible function application x square root of 1 plus 3 cos invisible function application x end root d x equals integral sin invisible function application x u to the power of 1 half end exponent fraction numerator d u over denominator negative 3 sin invisible function application x end fraction equals integral negative 1 third u to the power of 1 half end exponent d u end cell row blank cell equals negative 2 over 9 u to the power of 3 over 2 end exponent plus C equals negative 2 over 9 square root of left parenthesis 1 plus 3 cos invisible function application x right parenthesis cubed end root plus C end cell end table end style

أجد قيمة كل من التكاملات الآتية:

begin mathsize 20px style integral subscript 1 superscript 2 x squared over left parenthesis x cubed plus 1 right parenthesis squared d x end style (7)

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 subscript 1 superscript 2 x squared over left parenthesis x cubed plus 1 right parenthesis squared d x end cell row blank cell u equals x cubed plus 1 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 3 x squared not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 3 x squared end fraction end cell row blank cell x equals 2 not stretchy rightwards double arrow u equals 9 end cell row blank cell x equals 1 not stretchy rightwards double arrow u equals 2 end cell row blank cell integral subscript 1 superscript 2 x squared over left parenthesis x cubed plus 1 right parenthesis squared d x equals integral subscript 2 superscript 9 x squared over u squared fraction numerator d u over denominator 3 x squared end fraction equals integral subscript 2 superscript 9 1 third u to the power of negative 2 end exponent d u end cell row blank cell equals negative fraction numerator 1 over denominator 3 u end fraction vertical line subscript 2 superscript 9 equals negative 1 over 27 plus 1 over 6 equals 21 over 162 end cell end table end style

begin mathsize 20px style integral subscript 0 superscript 1 x square root of 3 x squared plus 2 end root d x end style (8)

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 subscript 0 superscript 1 x square root of 3 x squared plus 2 end root d x end cell row blank cell u equals 3 x squared plus 2 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 6 x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 6 x end fraction end cell row blank cell x equals 1 not stretchy rightwards double arrow u equals 5 end cell row blank cell x equals 0 not stretchy rightwards double arrow u equals 2 end cell 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 cell integral subscript 0 superscript 1 x square root of 3 x squared plus 2 end root d x equals integral subscript 2 superscript 5 x u to the power of 1 half end exponent fraction numerator d u over denominator 6 x end fraction end cell cell equals integral subscript 2 superscript 5 1 over 6 u to the power of 1 half end exponent d u end cell row blank cell equals 1 over 9 u to the power of 3 over 2 end exponent vertical line subscript 2 superscript 5 equals 1 over 9 square root of 125 minus 1 over 9 square root of 8 end cell end table end cell end table end style

begin mathsize 20px style integral subscript e superscript e squared end superscript fraction numerator left parenthesis ln invisible function application x right parenthesis squared over denominator x end fraction d x end style (9)

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 subscript e superscript e squared end superscript fraction numerator left parenthesis ln invisible function application x right parenthesis squared over denominator x end fraction d x end cell row blank cell u equals ln invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 1 over x not stretchy rightwards double arrow d x equals x d u end cell row blank cell x equals e not stretchy rightwards double arrow u equals 1 end cell row blank cell x equals e squared not stretchy rightwards double arrow u equals 2 end cell row blank cell integral subscript e superscript e squared end superscript fraction numerator left parenthesis ln invisible function application x right parenthesis squared over denominator x end fraction d x equals integral subscript 1 superscript 2 u squared over x x d u equals integral subscript 1 superscript 2 u squared d u end cell row blank cell equals 1 third u cubed vertical line subscript 1 superscript 2 equals 8 over 3 minus 1 third equals 7 over 3 end cell end table end style

begin mathsize 20px style integral subscript 0 superscript 1 left parenthesis x plus 1 right parenthesis left parenthesis x squared plus 2 x right parenthesis to the power of 5 d x end style (10)

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 subscript 0 superscript 1 left parenthesis x plus 1 right parenthesis left parenthesis x squared plus 2 x right parenthesis to the power of 5 d x end cell row blank cell u equals x squared plus 2 x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 2 x plus 2 not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 2 x plus 2 end fraction end cell row blank cell x equals 1 not stretchy rightwards double arrow u equals 3 end cell row blank cell x equals 0 not stretchy rightwards double arrow u equals 0 end cell row blank cell integral subscript 0 superscript 1 left parenthesis x plus 1 right parenthesis left parenthesis x squared plus 2 x right parenthesis to the power of 5 d x equals integral subscript 0 superscript 3 left parenthesis x plus 1 right parenthesis u to the power of 5 fraction numerator d u over denominator 2 x plus 2 end fraction equals integral subscript 0 superscript 3 1 half u to the power of 5 d u end cell row blank cell equals 1 over 12 u to the power of 6 vertical line subscript 0 superscript 3 equals 729 over 12 end cell end table end style

التمثيل البياني(11) أجد مساحة المنطقة المظللة في التمثيل البياني المجاور.

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 A equals integral subscript 0 superscript 2 x square root of x squared plus 2 end root d x end cell row blank cell u equals x squared plus 2 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 2 x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 2 x end fraction end cell row blank cell x equals 2 not stretchy rightwards double arrow u equals 6 end cell row blank cell x equals 0 not stretchy rightwards double arrow u equals 2 end cell row blank cell table attributes columnspacing 1em end attributes row cell integral subscript 0 superscript 2 x square root of x squared plus 2 end root d x equals integral subscript 2 superscript 6 x u to the power of 1 half end exponent fraction numerator d u over denominator 2 x end fraction equals integral subscript 2 superscript 6 1 half u to the power of 1 half end exponent d u end cell row cell equals 1 third u to the power of 3 over 2 end exponent vertical line subscript 2 superscript 6 equals 1 third square root of 216 minus 1 third square root of 8 end cell end table end cell end table end style

(12) الإيراد الحدي: يمثل الاقتران: begin mathsize 20px style R to the power of straight prime left parenthesis x right parenthesis equals 50 plus 3.5 x e to the power of negative 0.1 x squared end exponent end style الإيراد الحدي (بالدينار) لكل قطعة تباع من إنتاج إحدى الشركات، حيث begin mathsize 20px style x end styleعدد القطع المبيعة، وbegin mathsize 20px style R left parenthesis x right parenthesis end style إيراد بيع begin mathsize 20px style x end style قطعة بالدينار. أجد اقتران الإيراد begin mathsize 20px style R left parenthesis x right parenthesis end style، علماً بأن begin mathsize 20px style R left parenthesis 0 right parenthesis equals 0 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 R left parenthesis x right parenthesis equals integral left parenthesis 50 plus 3.5 x e to the power of negative 0.1 x squared end exponent right parenthesis d x equals integral 50 d x plus integral 3.5 x e to the power of negative 0.1 x squared end exponent d x end cell row blank cell space of 2em equals 50 x plus integral 3.5 x e to the power of negative 0.1 x squared end exponent d x end cell row blank cell table attributes columnalign right columnspacing 1em end attributes row cell u equals negative 0.1 x squared not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals negative 0.2 x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator negative 0.2 x end fraction end cell end table end cell 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 integral left parenthesis 50 plus 3.5 x e to the power of negative 0.1 x squared end exponent right parenthesis d x equals 50 x plus integral 3.5 x e to the power of u fraction numerator d u over denominator negative 0.2 x end fraction end cell row blank cell equals 50 x plus integral negative 17.5 e to the power of u d u end cell end table end cell row blank cell space of 2em equals 50 x minus 17.5 e to the power of negative 0.1 x squared end exponent plus C end cell row blank cell R left parenthesis 0 right parenthesis equals 0 not stretchy rightwards double arrow 0 minus 17.5 plus C equals 0 not stretchy rightwards double arrow C equals 17.5 end cell end table end style

يمثل الاقتران begin mathsize 20px style bold italic f to the power of bold prime bold left parenthesis bold italic x bold right parenthesis end style في كل مما يأتي ميل المماس لمنحنى الاقتران begin mathsize 20px style bold italic f bold left parenthesis bold italic x bold right parenthesis end style المار بالنقطة المعطاة، 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 2 x left parenthesis 4 x squared minus 10 right parenthesis squared semicolon left parenthesis 2 comma 10 right parenthesis 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 f left parenthesis x right parenthesis equals integral 2 x left parenthesis 4 x squared minus 10 right parenthesis squared d x end cell row blank cell u equals 4 x squared minus 10 not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals 8 x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 8 x end fraction end cell row blank cell table attributes columnspacing 1em end attributes row cell integral 2 x left parenthesis 4 x squared minus 10 right parenthesis squared d x equals integral 2 x u squared fraction numerator d u over denominator 8 x end fraction equals integral 1 fourth u squared d u equals 1 over 12 u cubed plus C end cell row cell f left parenthesis 2 right parenthesis equals 10 not stretchy rightwards double arrow 18 plus C equals 10 end cell row cell not stretchy rightwards double arrow C equals negative 8 end cell row cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals 1 over 12 left parenthesis 4 x squared minus 10 right parenthesis cubed minus 8 end cell end table end cell end table end style

begin mathsize 20px style f to the power of straight prime left parenthesis x right parenthesis equals x squared e to the power of negative 0.2 x cubed end exponent comma left parenthesis 0 comma 3 over 2 right parenthesis end style (14)

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 x squared e to the power of negative 0.2 x cubed end exponent d x end cell row blank cell u equals negative 0.2 x cubed not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals negative 0.6 x squared not stretchy rightwards double arrow d x equals fraction numerator d u over denominator negative 0.6 x squared end fraction end cell row blank cell integral x squared e to the power of negative 0.2 x cubed end exponent d x equals integral x squared e to the power of u fraction numerator d u over denominator negative 0.6 x squared end fraction equals integral e to the power of u fraction numerator d u over denominator negative 0.6 end fraction end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 5 over 3 e to the power of negative 0.2 x cubed end exponent plus C end cell row blank cell f left parenthesis 0 right parenthesis equals 3 over 2 not stretchy rightwards double arrow negative 5 over 3 e to the power of u d u equals negative 5 over 3 e to the power of negative 0.2 x cubed end exponent plus C end cell row blank cell not stretchy rightwards double arrow f left parenthesis x right parenthesis equals negative 5 over 3 e to the power of negative 0.2 x cubed end exponent plus 19 over 6 end cell end table end style

(15) يتحرك جسيم في مسار مستقيم، وتعطى سرعته المتجهة بالاقتران: begin mathsize 20px style v left parenthesis t right parenthesis equals fraction numerator t over denominator square root of t squared plus 1 end root end fraction end style، حيث begin mathsize 20px style t end style الزمن بالثواني، وbegin mathsize 20px style v end style سرعته المتجهة بالمتر لكل ثانية. إذا بدأ الجسيم حركته من نقطة الأصل، فأجد موقعه بعد begin mathsize 20px style t end style ثانية من بدء الحركة.

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

11 / 02 / 2023

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