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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:

Error converting from MathML to accessible text. (7)

12x2(x3+1)2dxu=x3+1dudx=3x2dx=du3x2x=2u=9x=1u=212x2(x3+1)2dx=29x2u2du3x2=2913u2du=13u|29=127+16=21162

01x3x2+2dx (8)

01x3x2+2dxu=3x2+2dudx=6xdx=du6xx=1u=5x=0u=201x3x2+2dx=25xu12du6x=2516u12du=19u32|25=19125198

ee2(lnx)2xdx (9)

ee2(lnx)2xdxu=lnxdudx=1xdx=xdux=eu=1x=e2u=2ee2(lnx)2xdx=12u2xxdu=12u2du=13u3|12=8313=73

01(x+1)(x2+2x)5dx (10)

01(x+1)(x2+2x)5dxu=x2+2xdudx=2x+2dx=du2x+2x=1u=3x=0u=001(x+1)(x2+2x)5dx=03(x+1)u5du2x+2=0312u5du=112u6|03=72912

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

A=02xx2+2dxu=x2+2dudx=2xdx=du2xx=2u=6x=0u=202xx2+2dx=26xu12du2x=2612u12du=13u32|26=13216138

(12) الإيراد الحدي: يمثل الاقتران: R(x)=50+3.5xe0.1x2 الإيراد الحدي (بالدينار) لكل قطعة تباع من إنتاج إحدى الشركات، حيث xعدد القطع المبيعة، وR(x) إيراد بيع x قطعة بالدينار. أجد اقتران الإيراد R(x)، علماً بأن R(0)=0

R(x)=(50+3.5xe0.1x2)dx=50dx+3.5xe0.1x2dx=50x+3.5xe0.1x2dxu=0.1x2dudx=0.2xdx=du0.2x(50+3.5xe0.1x2)dx=50x+3.5xeudu0.2x=50x+17.5eudu=50x17.5e0.1x2+CR(0)=0017.5+C=0C=17.5

يمثل الاقتران f(x) في كل مما يأتي ميل المماس لمنحنى الاقتران f(x) المار بالنقطة المعطاة، أستعمل المعلومات المعطاة لإيجاد قاعدة الاقتران f(x):

f(x)=2x(4x210)2;(2,10) (13)

f(x)=2x(4x210)2dxu=4x210dudx=8xdx=du8x2x(4x210)2dx=2xu2du8x=14u2du=112u3+Cf(2)=1018+C=10C=8f(x)=112(4x210)38

f(x)=x2e0.2x3,(0,32) (14)

f(x)=x2e0.2x3dxu=0.2x3dudx=0.6x2dx=du0.6x2x2e0.2x3dx=x2eudu0.6x2=eudu0.6f(x)=53e0.2x3+Cf(0)=3253eudu=53e0.2x3+Cf(x)=53e0.2x3+196

(15) يتحرك جسيم في مسار مستقيم، وتعطى سرعته المتجهة بالاقتران: v(t)=tt2+1، حيث t الزمن بالثواني، وv سرعته المتجهة بالمتر لكل ثانية. إذا بدأ الجسيم حركته من نقطة الأصل، فأجد موقعه بعد t ثانية من بدء الحركة.

s(t)=tt2+1dtu=t2+1dudt=2tdt=du2ttt2+1dt=u12du2t=12u12du=u12+C=t2+1+Cs(t)=t2+1+Cs(0)=01+C=0C=1s(t)=t2+11

إعداد : شبكة منهاجي التعليمية

11 / 02 / 2023

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