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Nilai ∫81​31​​x23​1+x1​​dx adalah ....

Pertanyaan

Nilai 8131x231+x1dx adalah ....

  1. 19

  2. 38

  3. 57

  4. 76

  5. 95

Pembahasan Soal:

Rumus cepat integral substitusi

(a)(b)ndx=ban+11(b)n+1+C

Berdasarkan rumus cepat tersebut, maka:

8131x231+x1dx========8131(3x2)(1+x1)21dxx23x221+11(1+x1)21+18131233(1+x1)238131(2(1+x1)1+x18131)(2(1+3)1+3(2(1+8)1+8))(242(293))(16+54)38 

Jadi, hasil dari inetgral tersebut adalah 38.

Oleh karena itu, jawaban yang benar adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Terakhir diupdate 12 September 2021

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Pembahasan Soal:

Ingat bahwa open vertical bar x close vertical bar equals square root of x squared end root. Kemudian, perhatikan perhitungan berikut!

integral 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 table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open vertical bar x close vertical bar end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 8 x squared plus 3 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x squared plus 3 end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table integral 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 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 cell open parentheses square root of x squared end root close parentheses end cell end table 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 cell open parentheses 8 x squared plus 3 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x squared plus 3 end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table equals integral 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 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 cell open parentheses 8 x squared plus 3 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of left parenthesis x squared right parenthesis times left parenthesis 4 x squared plus 3 right parenthesis end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 24 x cubed plus 9 x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x to the power of 4 plus 3 x squared end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 16 x cubed plus 6 x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x to the power of 4 plus 3 x squared end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table

Misal u equals 4 x to the power of 4 plus 3 x squared, maka didapat perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 4 x to the power of 4 plus 3 x squared end cell row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals cell 16 x cubed plus 6 x end cell row cell integral open parentheses fraction numerator straight d u over denominator straight d x end fraction close parentheses space straight d x end cell equals cell integral open parentheses 16 x cubed plus 6 x close parentheses space straight d x end cell row cell integral straight d u end cell equals cell integral open parentheses 16 x cubed plus 6 x close parentheses space straight d x end cell end table

Oleh karena itu, didapat perhitungan sebagai berikut.

integral 3 x vertical line x vertical line open parentheses 8 x squared plus 3 close parentheses square root of 4 x squared plus 3 end root space straight d x equals integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 16 x cubed plus 6 x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x to the power of 4 plus 3 x squared end root end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table equals integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of u end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank u end table equals integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell u to the power of 1 half end exponent end cell end table space table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank u end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table 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 cell 2 over 3 end cell end table 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 cell u to the power of 3 over 2 end exponent end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell u to the power of 3 over 2 end exponent end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 4 x to the power of 4 plus 3 x squared close parentheses to the power of 3 over 2 end exponent end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 4 x to the power of 4 plus 3 x squared close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 4 x to the power of 4 plus 3 x squared end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table

Dengan demikian, hasil dari integral 3 x vertical line x vertical line left parenthesis 8 x squared plus 3 right parenthesis square root of 4 x squared plus 3 end root space straight d x adalah open parentheses 4 x to the power of 4 plus 3 x squared close parentheses square root of 4 x to the power of 4 plus 3 x squared end root plus C.

Jadi, jawaban yang tepat adalah C.

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Roboguru

Hasil dari  adalah ....

Pembahasan Soal:

Dari soal diberikan bentuk integral integral left parenthesis 5 x plus 3 right parenthesis to the power of 14 blank d x.

Misal 5 x plus 3 equals u, maka didapat bentuk sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d u over denominator d x end fraction end cell equals 5 row cell integral fraction numerator d u over denominator d x end fraction space d x end cell equals cell integral 5 space d x end cell row cell integral d u end cell equals cell integral 5 space d x end cell row cell 1 fifth integral d u end cell equals cell integral d x end cell end table

Selanjutnya, perhatikan perhitungan berikut!

table attributes columnalign right center left columnspacing 0px end attributes row cell integral left parenthesis 5 x plus 3 right parenthesis to the power of 14 space d x end cell equals cell 1 fifth integral u to the power of 14 blank d u end cell row blank equals cell 1 fifth times 1 over 15 u to the power of 15 plus C end cell row blank equals cell 1 over 75 u to the power of 15 plus C end cell end table

Ingat kembali bahwa u equals 5 x plus 3 sehingga diperoleh hasil integral sebagai berikut.

begin mathsize 14px style 1 over 75 u to the power of 15 plus C equals 1 over 75 left parenthesis 5 x plus 3 right parenthesis to the power of 15 plus C end style

Jadi, jawaban yang tepat adalah D.space space

0

Roboguru

Hasil dari  adalah ....

Pembahasan Soal:

Dari soal diberikan bentuk integral undefined.

Misal u equals x squared minus 7 x plus 9, maka didapat bentuk sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d u over denominator d x end fraction end cell equals cell 2 straight x minus 7 end cell row cell integral open parentheses fraction numerator d u over denominator d x end fraction close parentheses space d x end cell equals cell integral open parentheses 2 straight x minus 7 close parentheses space d x end cell row cell integral d u end cell equals cell integral open parentheses 2 straight x minus 7 close parentheses space d x end cell end table

Selanjutnya, perhatikan perhitungan berikut!

 table attributes columnalign right center left columnspacing 0px end attributes row cell integral open parentheses 2 x minus 7 close parentheses open parentheses x squared minus 7 x plus 9 close parentheses to the power of 8 blank d x end cell equals cell integral u to the power of 8 blank d u end cell row blank equals cell 1 over 9 u to the power of 9 plus C end cell end table

Ingat kembali bahwa u equals x squared minus 7 x plus 9 sehingga diperoleh hasil integral sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral open parentheses 2 x minus 7 close parentheses open parentheses x squared minus 7 x plus 9 close parentheses to the power of 8 blank d x end cell row blank equals cell 1 over 9 u to the power of 9 plus C end cell row blank equals cell 1 over 9 open parentheses x squared minus 7 x plus 9 close parentheses to the power of 9 plus C end cell end table

Jadi, jawaban yang tepat adalah A.

0

Roboguru

Hasil dari  adalah ….

Pembahasan Soal:

Dari soal diberikan integral aljabar sebagai berikut.

integral fraction numerator 9 x squared minus 60 x plus 12 over denominator square root of x cubed minus 10 x squared plus 4 x minus 7 end root end fraction blank straight d x

 

Jika x cubed minus 10 x squared plus 4 x minus 7 equals u, maka didapat perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals cell 3 x squared minus 20 x plus 4 end cell row cell integral open parentheses fraction numerator straight d u over denominator straight d x end fraction close parentheses straight d x end cell equals cell integral open parentheses 3 x squared minus 20 x plus 4 close parentheses straight d x end cell row cell integral straight d u end cell equals cell integral open parentheses 3 x squared minus 20 x plus 4 close parentheses straight d x end cell end table  

Kemudian, perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral fraction numerator 9 x squared minus 60 x plus 12 over denominator square root of x cubed minus 10 x squared plus 4 x minus 7 end root end fraction blank straight d x end cell row blank equals cell integral fraction numerator 3 open parentheses 3 x squared minus 20 x plus 4 close parentheses over denominator square root of x cubed minus 10 x squared plus 4 x minus 7 end root end fraction blank straight d x end cell row blank equals cell 3 integral fraction numerator 1 over denominator square root of u end fraction blank straight d u end cell row blank equals cell 3 integral u to the power of negative 1 half end exponent blank straight d u end cell row blank equals cell 3 times 2 u to the power of 1 half end exponent plus straight C end cell row blank equals cell 6 square root of u plus straight C end cell end table   

Ingat kembali bahwa u equals x cubed minus 10 x squared plus 4 x minus 7 sehingga diperoleh hasil sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 9 x squared minus 60 x plus 12 over denominator square root of x cubed minus 10 x squared plus 4 x minus 7 end root end fraction blank straight d x end cell equals cell 6 square root of u plus C end cell row blank equals cell 6 square root of x cubed minus 10 x squared plus 4 x minus 7 end root plus C end cell end table

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Hasil dari  adalah ….

Pembahasan Soal:

Dari soal diberikan integral aljabar sebagai berikut.

integral fraction numerator 3 x squared minus 8 x plus 7 over denominator open parentheses x cubed minus 4 x squared plus 7 x minus 15 close parentheses cubed end fraction blank straight d x

Jika x cubed minus 4 x squared plus 7 x minus 15 equals u, maka didapat perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals cell 3 x squared minus 8 x plus 7 end cell row cell integral open parentheses fraction numerator straight d u over denominator straight d x end fraction close parentheses straight d x end cell equals cell integral open parentheses 3 x squared minus 8 x plus 7 close parentheses straight d x end cell row cell integral straight d u end cell equals cell integral open parentheses 3 x squared minus 8 x plus 7 close parentheses straight d x end cell end table 

Kemudian, perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 3 x squared minus 8 x plus 7 over denominator open parentheses x cubed minus 4 x squared plus 7 x minus 15 close parentheses cubed end fraction blank straight d x end cell equals cell integral 1 over u cubed blank straight d u end cell row blank equals cell integral u to the power of negative 3 end exponent blank straight d u end cell row blank equals cell negative 1 half u to the power of negative 2 end exponent plus C end cell row blank equals cell negative fraction numerator 1 over denominator 2 u squared end fraction plus C end cell end table 

Ingat kembali bahwa u equals x cubed minus 4 x squared plus 7 x minus 15 sehingga diperoleh hasil sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 3 x squared minus 8 x plus 7 over denominator open parentheses x cubed minus 4 x squared plus 7 x minus 15 close parentheses cubed end fraction blank straight d x end cell equals cell negative fraction numerator 1 over denominator 2 u squared end fraction plus C end cell row blank equals cell negative fraction numerator 1 over denominator 2 open parentheses x cubed minus 4 x squared plus 7 x minus 15 close parentheses squared end fraction plus C end cell end table 

Jadi, jawaban yang tepat adalah E.

1

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