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Hasil dari integral 3 x square root of 3 x squared plus 1 end root straight d x adalah ...

  1. 1 half open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C

  2. 2 over 3 open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C

  3. negative 2 over 3 open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C

  4. 1 third open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C

  5. negative 1 half open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C

Pembahasan Soal:

Perhatikan perhitungan berikut.

Ingat, integral tak tentu:

integral x to the power of n straight d x equals fraction numerator 1 over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus C

Misalkan:

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 3 x squared plus 1 end cell row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals cell 6 x end cell row cell straight d u end cell equals cell 6 x straight d x end cell row cell 1 half straight d u end cell equals cell 3 x straight d x end cell end table

Maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral 3 x square root of 3 x squared plus 1 end root straight d x end cell equals cell integral square root of 3 x squared plus 1 end root 3 x straight d x end cell row blank equals cell integral square root of u 1 half straight d u end cell row blank equals cell 1 half integral square root of u straight d u end cell row blank equals cell 1 half integral open parentheses u close parentheses to the power of 1 half end exponent straight d u end cell row blank equals cell 1 half open square brackets fraction numerator 1 over denominator 1 plus begin display style 1 half end style end fraction u to the power of 1 half plus 1 end exponent close square brackets plus C end cell row blank equals cell 1 half open square brackets fraction numerator 1 over denominator begin display style 2 over 2 end style plus begin display style 1 half end style end fraction u to the power of 1 half plus 2 over 2 end exponent close square brackets plus C end cell row blank equals cell 1 half open square brackets fraction numerator 1 over denominator begin display style 3 over 2 end style end fraction u to the power of 3 over 2 end exponent close square brackets plus C end cell row blank equals cell 1 half open square brackets 2 over 3 square root of u cubed end root close square brackets plus C end cell row blank equals cell 1 third u square root of u plus C end cell row blank equals cell 1 third open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C end cell end table

Jadi, hasil dari integral 3 x square root of 3 x squared plus 1 end root straight d x adalah 1 third open parentheses 3 x squared plus 1 close parentheses square root of 3 x squared plus 1 end root plus C.

Dengan demikian, jawaban yang tepat adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Ayu

Mahasiswa/Alumni Universitas Negeri Padang

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Misalkan space straight u equals 4 straight x squared plus 3  Akibatnya space du equals 8 straight x space dx space atau space 1 fourth du equals 2 straight x space dx  integral 2 straight x open parentheses 4 straight x squared plus 3 close parentheses to the power of 3 over 2 end exponent dx equals integral open parentheses 4 straight x squared plus 3 close parentheses to the power of 3 over 2 end exponent 2 straight x space dx  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 space space space space space space equals integral straight space straight u to the power of 3 over 2 end exponent open parentheses 1 fourth du 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 space space space space space space equals 1 fourth integral straight space straight u to the power of 3 over 2 end exponent du  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 space space space space space space equals 1 fourth straight space.2 over 5 straight space straight u to the power of 5 over 2 end exponent plus straight C  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 space space space space space space equals 1 over 10 straight space straight u squared straight space. straight space square root of straight u plus straight C  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 space space space space space space equals 1 over 10 straight space open parentheses 4 straight x squared plus 3 close parentheses squared square root of 4 straight x squared plus 3 end root plus straight C

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Pembahasan Soal:

M i s a l space u space equals x squared minus x plus 5 comma space m a k a space d u equals left parenthesis 2 x minus 1 right parenthesis d x. space S e h i n g g a comma  integral subscript space superscript space left parenthesis 2 x minus 1 right parenthesis square root of x squared minus x plus 5 end root space d x equals space integral subscript space superscript space square root of u space d u space  equals fraction numerator 1 over denominator 1 half plus 1 end fraction u to the power of 1 half plus 1 end exponent plus C  equals 2 over 3 u square root of u plus C  equals 2 over 3 left parenthesis x squared minus x plus 5 right parenthesis square root of x squared minus x plus 5 end root plus C

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begin mathsize 14px style integral open parentheses 3 x plus 2 close parentheses open parentheses 3 x squared plus 4 x minus 1 close parentheses squared d x  misalkan colon space u equals 3 x squared plus 4 x minus 1 fraction numerator d u over denominator d x end fraction equals 6 x plus 4 1 half fraction numerator d u over denominator d x end fraction equals 3 x plus 2 1 half straight d u equals 3 x plus 2 space straight d x  integral open parentheses 3 x plus 2 close parentheses open parentheses 3 x squared plus 4 x minus 1 close parentheses squared d x equals integral 1 half space u squared space straight d u equals 1 half times fraction numerator 1 over denominator 2 plus 1 end fraction u cubed plus straight C equals 1 over 6 u cubed plus straight C equals 1 over 6 left parenthesis 3 x squared plus 4 x minus 1 right parenthesis cubed plus straight C space space end style  

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Pembahasan Soal:

Misalkan u equals 3 x squared minus 4 x

Akibatnya d u equals 6 x minus 4 space d x space atau space 1 half d u equals 3 x minus 2 space d x

integral left parenthesis 3 x minus 2 right parenthesis square root of 3 x squared minus 4 x end root space d x  equals integral square root of 3 x squared minus 4 x end root left parenthesis 3 x minus 2 right parenthesis d x  equals integral square root of u space open parentheses 1 half d u close parentheses  equals 1 half integral u to the power of 1 half end exponent space d u  equals 1 half.2 over 3 u to the power of 3 over 2 end exponent plus C  equals 1 third u. square root of u plus C  equals 1 third left parenthesis 3 x squared minus 4 x right parenthesis square root of 3 x squared minus 4 x end root plus C

 

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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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