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Pertanyaan

Hasil dari integral x open parentheses x plus 4 close parentheses to the power of 5 space text dx end text adalah ...

  1. 1 third x open parentheses x plus 4 close parentheses to the power of 6 minus 1 over 42 open parentheses x plus 4 close parentheses to the power of 7 plus C 

  2. 1 third x open parentheses x plus 4 close parentheses to the power of 6 plus 1 over 42 open parentheses x plus 4 close parentheses to the power of 7 plus C 

  3. 1 over 6 x open parentheses x plus 4 close parentheses to the power of 6 minus 1 over 42 open parentheses x plus 4 close parentheses to the power of 7 plus C 

  4. 1 over 6 x open parentheses x plus 4 close parentheses to the power of 6 plus 1 over 21 open parentheses x plus 4 close parentheses to the power of 7 plus C 

  5. 1 over 6 x open parentheses x plus 4 close parentheses to the power of 6 minus 1 over 21 open parentheses x plus 4 close parentheses to the power of 7 plus C 

Pembahasan Soal:

Peyelesaian integral tersebut menggunakan metode integral parsial sebagai berikut.

integral x open parentheses x plus 4 close parentheses to the power of 5 space d x equals 1 over 6 x open parentheses x plus 4 close parentheses to the power of 6 minus 1 over 42 open parentheses x plus 4 close parentheses to the power of 7 plus C

Oleh karena itu, jawaban yang tepat adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Eka

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 07 Juni 2021

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Pertanyaan yang serupa

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Pembahasan Soal:

begin mathsize 14px style integral 2 x open parentheses 5 minus x close parentheses cubed d x equals integral u d v  u equals 2 x space  d u equals 2 space d x    d v equals open parentheses 5 minus x close parentheses cubed space d x  v equals integral open parentheses 5 minus x close parentheses cubed d x equals negative 1 fourth open parentheses 5 minus x close parentheses to the power of 4    integral u d v equals u. v minus integral v d u  equals 2 x open parentheses negative 1 fourth open parentheses 5 minus x close parentheses to the power of 4 close parentheses minus integral negative 1 fourth open parentheses 5 minus x close parentheses to the power of 4.2 space d x  equals negative 1 half x open parentheses 5 minus x close parentheses to the power of 4 plus 1 half open parentheses negative 1 fifth open parentheses 5 minus x close parentheses to the power of 5 close parentheses plus c  equals negative 1 half x open parentheses 5 minus x close parentheses to the power of 4 minus 1 over 10 open parentheses 5 minus x close parentheses to the power of 5 plus c  equals negative 1 over 10 open parentheses 5 minus x close parentheses to the power of 4 open parentheses 5 x plus open parentheses 5 minus x close parentheses close parentheses space plus c  equals negative 1 over 10 open parentheses 4 x plus 5 close parentheses open parentheses 5 minus x close parentheses to the power of 4 space plus c end style

1

Roboguru

Selesaikan setiap integral berikut dengan cara manipulasi aljabar dan substitusi. a.

Pembahasan Soal:

0

Roboguru

Tentukan integral berikut dengan integral parsial  b.

Pembahasan Soal:

Misalkan u equals x cubed dan d v equals open parentheses x plus 3 close parentheses cubed maka:

table row cell u equals x cubed end cell rightwards arrow cell fraction numerator d u over denominator d x end fraction equals 3 x squared end cell row blank blank cell d u equals 3 x squared space d x end cell row cell d v equals open parentheses x plus 1 close parentheses cubed end cell rightwards arrow cell v equals integral open parentheses x plus 1 close parentheses cubed space d x end cell row blank blank cell v equals 1 fourth open parentheses x plus 1 close parentheses to the power of 4 space end cell end table

Dengan menggunakan konsep integral parsial maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral x cubed open parentheses x plus 1 close parentheses cubed d x end cell equals cell u times v minus integral v space d u end cell row blank equals cell x cubed times 1 fourth open parentheses x plus 1 close parentheses cubed minus integral 1 fourth open parentheses x plus 1 close parentheses to the power of 4 times 3 x squared space d x end cell row blank equals cell 1 fourth x cubed open parentheses x plus 1 close parentheses to the power of 4 minus 1 fourth integral open parentheses x plus 1 close parentheses to the power of 4 times 3 x squared space d x end cell end table  

Kemudian tentukan integral dari table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x plus 1 close parentheses end cell end table to the power of 4 table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table to the power of 2 space end exponent table attributes columnalign right center left columnspacing 0px end attributes row blank blank d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table, ikuti langkah langkah di atas sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell u equals 3 x squared end cell rightwards arrow cell d u equals 6 x space d x end cell row cell d v equals open parentheses x plus 1 close parentheses to the power of 4 end cell rightwards arrow cell v equals integral open parentheses x plus 1 close parentheses to the power of 4 space d x end cell row blank blank cell v equals 1 fifth open parentheses x plus 1 close parentheses to the power of 5 plus C end cell end table end cell end table  

maka:

  table attributes columnalign right center left columnspacing 0px end attributes row cell integral open parentheses x plus 1 close parentheses to the power of 4 3 x squared space d x end cell equals cell u times v minus integral v space d u end cell row blank equals cell 3 x squared times 1 fifth open parentheses x plus 1 close parentheses to the power of 5 minus integral 1 fifth open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x end cell row blank equals cell 1 fifth 3 x squared open parentheses x plus 1 close parentheses to the power of 5 minus 1 fifth integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x end cell end table

Kemudian tentukan integral dari integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes rowalign bottom baseline baseline end attributes row cell u equals 6 x end cell rightwards arrow cell d u equals 6 space d x end cell row cell d v equals open parentheses x plus 1 close parentheses to the power of 5 end cell rightwards arrow cell v equals integral open parentheses x plus 1 close parentheses to the power of 5 space d x end cell row blank blank cell v equals 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 plus C end cell end table end cell end table 

maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x end cell equals cell u times v minus integral v space d u end cell row blank equals cell 6 x times 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 minus integral 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 times 6 d x end cell row blank equals cell 1 over 6 6 x open parentheses x plus 1 close parentheses to the power of 6 minus integral open parentheses x plus 1 close parentheses to the power of 6 d x end cell row blank equals cell 1 over 6 6 x open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 7 open parentheses x plus 1 close parentheses to the power of 7 plus C end cell end table  

kemudian substitusikan ke table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x plus 1 close parentheses end cell end table to the power of 4 table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table to the power of 2 space end exponent table attributes columnalign right center left columnspacing 0px end attributes row blank blank d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table maka:

begin mathsize 12px style integral x cubed open parentheses x plus 1 close parentheses cubed space d x equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus 1 fourth integral open parentheses x plus 1 close parentheses to the power of 4 times 3 x squared space d x equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus 1 fourth open square brackets fraction numerator 3 x squared over denominator 5 end fraction open parentheses x plus 1 close parentheses to the power of 5 minus 1 fifth integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x close square brackets equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus fraction numerator 3 x squared over denominator 20 end fraction open parentheses x plus 1 close parentheses to the power of 5 plus 1 over 20 open square brackets fraction numerator 6 x over denominator 6 end fraction open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 7 open parentheses x plus 1 close parentheses to the power of 7 close square brackets equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus fraction numerator 3 x squared over denominator 20 end fraction open parentheses x plus 1 close parentheses to the power of 5 plus x over 20 open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 140 open parentheses x plus 1 close parentheses to the power of 7 end style  

Dengan demikian hasil dari  integral x cubed open parentheses x plus 1 close parentheses cubed space d x adalah

size 12px x to the power of size 12px 3 over size 12px 4 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 4 size 12px minus fraction numerator size 12px 3 size 12px x to the power of size 12px 2 over denominator size 12px 20 end fraction begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 5 size 12px plus size 12px x over size 12px 20 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 6 size 12px minus size 12px 1 over size 12px 140 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 7 

0

Roboguru

Selesaikan setiap integral parsial di bawah ini dengan cara tabulasi. b.

Pembahasan Soal:

0

Roboguru

Pembahasan Soal:

nilai integral x open parentheses x minus 6 close parentheses squared d x dapat ditentukan menggunakan cara parsial yaitu integral u d v equals u v minus integral v d u dengan u equals x space text dan end text space d v equals open parentheses x minus 6 close parentheses squared d x comma maka

u space space equals x space space space space space space space text dan end text space d v equals open parentheses x minus 6 close parentheses squared d x d u equals 1 d x space space space space space space space space space space space space v equals integral open parentheses x minus 6 close parentheses squared d x d u equals d x space space space space space space space space space space space space space space v equals integral open parentheses x minus 6 close parentheses squared d open parentheses x minus 6 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space v equals open parentheses x minus 6 close parentheses cubed over 3 plus text C end text

Nilai integral x open parentheses x minus 6 close parentheses squared d x adalah

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral u d v end cell equals cell u v minus integral v d u end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell open parentheses x cross times open parentheses x minus 6 close parentheses cubed over 3 close parentheses minus integral open parentheses x minus 6 close parentheses cubed over 3 d x end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x open parentheses x minus 6 close parentheses cubed over denominator 3 end fraction minus 1 third integral open parentheses x minus 6 close parentheses cubed d open parentheses x minus 6 close parentheses end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x open parentheses x minus 6 close parentheses cubed over denominator 3 end fraction minus 1 third open parentheses x minus 6 close parentheses to the power of 4 over 4 plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell open parentheses x minus 6 close parentheses squared over 3 open parentheses x open parentheses x minus 6 close parentheses minus 1 fourth open parentheses x minus 6 close parentheses squared close parentheses plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x squared minus 12 x plus 36 over denominator 3 end fraction open parentheses x squared minus 6 x minus 1 fourth open parentheses x squared minus 12 x plus 36 close parentheses close parentheses plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x squared minus 12 x plus 36 over denominator 3 end fraction open parentheses fraction numerator 4 x squared minus 24 x minus x squared plus 12 x minus 36 over denominator 4 end fraction close parentheses plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x squared minus 12 x plus 36 over denominator up diagonal strike 3 end fraction open parentheses fraction numerator up diagonal strike 3 x squared begin display style stack up diagonal strike negative 12 end strike with 4 on top end style x begin display style stack up diagonal strike negative 36 end strike with 12 on top end style over denominator 4 end fraction close parentheses plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator open parentheses x squared minus 12 x plus 36 close parentheses open parentheses x squared minus 4 x minus 12 close parentheses over denominator 4 end fraction plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell fraction numerator x to the power of 4 minus 16 x cubed plus 72 x squared minus 432 over denominator 4 end fraction plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell 1 fourth x to the power of 4 minus 4 x cubed plus 18 x squared minus 108 plus text C end text end cell row cell integral x open parentheses x minus 6 close parentheses squared d x end cell equals cell 1 fourth x to the power of 4 minus 4 x cubed plus 18 x squared plus text C end text end cell end table end style


Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

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