Roboguru

Tentukan integral tak tentu berikut: g.

Pertanyaan

Tentukan integral tak tentu berikut:

g. integral open parentheses 2 x minus 1 over x close parentheses squared space d x

Pembahasan Soal:

Rumus dasar integral yaitu:

  • integral k x to the power of n space d x equals fraction numerator k over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus straight C dengan sayarat straight n not equal to negative 1
  • integral k space d x equals k x plus straight C, suatu konstanta

Diperoleh penyelesaiannya yaitu:

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

Dengan demikian, integral tak tentu dari integral open parentheses 2 x minus 1 over x close parentheses squared space d x adalah Error converting from MathML to accessible text..

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 11 Juli 2021

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Pertanyaan yang serupa

Hasil dari  adalah ....

Pembahasan Soal:

Untuk menentukan hasil integral dari integral open parentheses 2 minus x close parentheses squared open parentheses 3 x plus 1 close parentheses space d x, perhatikan perhitungan berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral open parentheses 2 minus x close parentheses squared open parentheses 3 x plus 1 close parentheses space d x end cell row blank equals cell integral open parentheses 3 x cubed minus 11 x squared plus 8 x plus 4 close parentheses blank d x end cell row blank equals cell integral 3 x cubed blank d x minus integral 11 x squared blank d x plus integral 8 x blank d x plus integral 4 blank d x end cell row blank equals cell 3 open parentheses fraction numerator 1 over denominator 3 plus 1 end fraction x to the power of 3 plus 1 end exponent plus C subscript 1 close parentheses minus 11 open parentheses fraction numerator 1 over denominator 2 plus 1 end fraction x to the power of 2 plus 1 end exponent plus C subscript 2 close parentheses plus end cell row blank blank cell 8 open parentheses fraction numerator 1 over denominator 1 plus 1 end fraction x to the power of 1 plus 1 end exponent plus C subscript 3 close parentheses plus 4 open parentheses x plus C subscript 4 close parentheses end cell row blank equals cell 3 open parentheses 1 fourth x to the power of 4 plus C subscript 1 close parentheses minus 11 open parentheses 1 third x cubed plus C subscript 2 close parentheses plus end cell row blank blank cell 8 open parentheses 1 half x squared plus C subscript 3 close parentheses plus 4 open parentheses x plus C subscript 4 close parentheses end cell row blank equals cell 3 over 4 x to the power of 4 plus 3 C subscript 1 minus 11 over 3 x cubed minus 11 C subscript 2 plus 4 x squared plus 8 C subscript 3 plus 4 x plus 4 C subscript 4 end cell row blank equals cell 3 over 4 x to the power of 4 minus 11 over 3 x cubed plus 4 x squared plus 4 x plus 3 C subscript 1 minus 11 C subscript 2 plus 8 C subscript 3 plus 4 C subscript 4 end cell end table end style

Misalkan 3 C subscript 1 minus 11 C subscript 2 plus 8 C subscript 3 plus 4 C subscript 4 equals C, maka didapat hasil perhitungan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 4 x to the power of 4 minus 11 over 3 x cubed plus 4 x squared plus 4 x plus 3 C subscript 1 minus 11 C subscript 2 plus 8 C subscript 3 plus 4 C subscript 4 end cell row blank equals cell 3 over 4 x to the power of 4 minus 11 over 3 x cubed plus 4 x squared plus 4 x plus C end cell end table end style

Jadi, jawaban yang tepat adalah A.

0

Roboguru

Hasil dari  adalah ....

Pembahasan Soal:

Perhatikan perhitungan berikut!

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral 1 half open parentheses x plus 1 close parentheses open parentheses 1 minus x close parentheses squared blank straight d x end cell row blank equals cell integral 1 half open parentheses x cubed minus x squared minus x plus 1 close parentheses blank straight d x end cell row blank equals cell 1 half integral open parentheses x cubed minus x squared minus x plus 1 close parentheses blank straight d x end cell row blank equals cell 1 half open parentheses integral x cubed blank straight d x minus integral x squared blank straight d x minus integral x blank straight d x plus integral 1 blank straight d x close parentheses end cell row blank equals cell 1 half open parentheses open parentheses fraction numerator 1 over denominator 3 plus 1 end fraction x to the power of 3 plus 1 end exponent plus C subscript 1 close parentheses minus open parentheses fraction numerator 1 over denominator 2 plus 1 end fraction x to the power of 2 plus 1 end exponent plus C subscript 2 close parentheses minus open parentheses fraction numerator 1 over denominator 1 plus 1 end fraction x to the power of 1 plus 1 end exponent plus C subscript 3 close parentheses plus open parentheses x plus C subscript 4 close parentheses close parentheses end cell row blank equals cell 1 half open parentheses open parentheses 1 fourth x to the power of 4 plus C subscript 1 close parentheses minus open parentheses 1 third x cubed plus C subscript 2 close parentheses minus open parentheses 1 half x squared plus C subscript 3 close parentheses plus open parentheses x plus C subscript 4 close parentheses close parentheses end cell row blank equals cell 1 half open parentheses 1 fourth x to the power of 4 plus C subscript 1 minus 1 third x cubed minus C subscript 2 minus 1 half x squared minus C subscript 3 plus x plus C subscript 4 close parentheses end cell row blank equals cell 1 over 8 x to the power of 4 plus C subscript 1 over 2 minus 1 over 6 x cubed minus C subscript 2 over 2 minus 1 fourth x squared minus C subscript 3 over 2 plus x over 2 plus C subscript 4 over 2 end cell row blank equals cell 1 over 8 x to the power of 4 minus 1 over 6 x cubed minus 1 fourth x squared plus x over 2 plus C subscript 1 over 2 minus C subscript 2 over 2 minus C subscript 3 over 2 plus C subscript 4 over 2 end cell end table end style 

Jika begin mathsize 12px style C subscript 1 over 2 minus C subscript 2 over 2 minus C subscript 3 over 2 plus C subscript 4 over 2 equals C comma end style maka diperoleh hasil sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 over 8 x to the power of 4 minus 1 over 6 x cubed minus 1 fourth x squared plus x over 2 plus C subscript 1 over 2 minus C subscript 2 over 2 minus C subscript 3 over 2 plus C subscript 4 over 2 end cell row blank equals cell 1 over 8 x to the power of 4 minus 1 over 6 x cubed minus 1 fourth x squared plus x over 2 plus C end cell end table 

Jadi, jawaban yang tepat adalah B.

1

Roboguru

Jika , hasil dari  adalah ...

Pembahasan Soal:

Ingat kembali:

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

Diketahui fungsi f left parenthesis x right parenthesis equals left parenthesis 3 plus x right parenthesis left parenthesis 3 x minus 1 right parenthesis, maka integral f left parenthesis x right parenthesis space d x dapat ditentukan sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral f left parenthesis x right parenthesis space d x space end cell equals cell integral left parenthesis x plus 3 right parenthesis left parenthesis 3 x minus 1 right parenthesis space d x end cell row blank equals cell integral open parentheses 3 x squared minus x plus 9 x minus 3 close parentheses space d x end cell row blank equals cell integral open parentheses 3 x squared plus 8 x minus 3 close parentheses space d x end cell row blank equals cell integral open parentheses 3 x squared plus 8 x to the power of 1 minus 3 x to the power of 0 close parentheses space d x end cell row blank equals cell fraction numerator 3 over denominator 2 plus 1 end fraction x to the power of 2 plus 1 end exponent plus fraction numerator 8 over denominator 1 plus 1 end fraction x to the power of 1 plus 1 end exponent minus fraction numerator 3 over denominator 0 plus 1 end fraction x to the power of 0 plus 1 end exponent plus C end cell row blank equals cell 3 over 3 x cubed plus 8 over 2 x squared minus 3 x plus C end cell row blank equals cell x cubed plus 4 x squared minus 3 x plus C end cell end table 

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Hasil dari  adalah ...

Pembahasan Soal:

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell integral a x to the power of n space straight d x end cell equals cell fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus C end cell row cell integral open parentheses f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses close parentheses space straight d x end cell equals cell integral f open parentheses x close parentheses space straight d x plus-or-minus integral g open parentheses x close parentheses space straight d x end cell row cell integral k times f open parentheses x close parentheses space straight d x end cell equals cell k integral f open parentheses x close parentheses space straight d x end cell end table   

Maka, hasil dari integral fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses over denominator square root of x end fraction space straight d x adalah:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses over denominator square root of x end fraction space straight d x end cell row blank equals cell integral x to the power of negative 1 half end exponent open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses space straight d x end cell row blank equals cell integral open parentheses 2 x to the power of 1 half end exponent minus x to the power of negative 1 half end exponent close parentheses open parentheses x plus 2 close parentheses space straight d x end cell row blank equals cell integral 2 x to the power of 3 over 2 end exponent plus 3 x to the power of 1 half end exponent minus 2 x to the power of negative 1 half end exponent space d x end cell row blank equals cell 2 integral x to the power of 3 over 2 end exponent space d x plus 3 integral x to the power of 1 half end exponent space d x minus 2 integral x to the power of negative 1 half end exponent space d x end cell row blank equals cell fraction numerator 2 over denominator begin display style bevelled 5 over 2 end style end fraction x to the power of 5 over 2 end exponent plus fraction numerator 3 over denominator begin display style bevelled 3 over 2 end style end fraction x to the power of 3 over 2 end exponent minus fraction numerator 2 over denominator begin display style bevelled 1 half end style end fraction x to the power of 1 half end exponent plus C end cell row blank equals cell 4 over 5 x squared square root of x plus 2 x square root of x minus 4 square root of x plus C end cell end table  

Jadi, integral fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x plus 2 close parentheses over denominator square root of x end fraction space straight d x equals 4 over 5 x squared square root of x plus 2 x square root of x minus 4 square root of x plus C.

0

Roboguru

Selesaikanlah integral berikut! a.

Pembahasan Soal:

Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell integral x open parentheses square root of x minus 1 close parentheses space straight d x end cell equals cell space integral x open parentheses x to the power of 1 half end exponent minus 1 close parentheses space straight d x end cell row blank equals cell space integral x to the power of 3 over 2 end exponent minus x space straight d x end cell row blank equals cell fraction numerator 1 over denominator begin display style 5 over 2 end style end fraction x to the power of 5 over 2 end exponent minus 1 half x squared plus C end cell row blank equals cell 2 over 5 x to the power of 5 over 2 end exponent minus 1 half x squared plus C end cell row blank equals cell 2 over 5 x squared square root of x minus 1 half x squared plus C end cell end table 

Jadi, Error converting from MathML to accessible text..

1

Roboguru

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Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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