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Tentukan integral tak tentu berikut: f.

Pertanyaan

Tentukan integral tak tentu berikut:

f. integral open parentheses 3 square root of x minus x square root of x plus 2 close parentheses 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 blank blank cell integral open parentheses 3 square root of x minus x square root of x plus 2 close parentheses d x end cell row blank equals cell integral 3 square root of x space d x minus integral x square root of x space d x plus integral 2 space d x end cell row blank equals cell integral 3 x to the power of 1 half end exponent space d x minus integral x x to the power of 1 half end exponent space d x plus integral 2 space d x end cell row blank equals cell integral 3 x to the power of 1 half end exponent space d x minus integral x to the power of 1 plus 1 half end exponent space space d x plus integral 2 space d x end cell row blank equals cell integral 3 x to the power of 1 half end exponent space d x minus integral x to the power of 3 over 2 end exponent space space d x plus integral 2 space d x end cell row blank equals cell fraction numerator 3 over denominator begin display style 1 half end style plus 1 end fraction x to the power of 1 half plus 1 end exponent minus fraction numerator 1 over denominator begin display style 3 over 2 end style plus 1 end fraction x to the power of 3 over 2 plus 1 end exponent plus 2 x plus straight C end cell row blank equals cell fraction numerator 3 over denominator begin display style 1 half end style plus begin display style 2 over 2 end style end fraction straight x to the power of 1 half plus 2 over 2 end exponent minus fraction numerator 1 over denominator begin display style 3 over 2 end style plus begin display style 2 over 2 end style end fraction x to the power of 3 over 2 plus 2 over 2 end exponent plus 2 x plus straight C end cell row blank equals cell fraction numerator 3 over denominator begin display style 3 over 2 end style end fraction x to the power of 3 over 2 end exponent minus 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 plus 2 x plus straight C end cell row blank equals cell down diagonal strike 3 times fraction numerator 2 over denominator down diagonal strike 3 end fraction x to the power of 3 over 2 end exponent minus 1 times 2 over 5 x to the power of 5 over 2 end exponent plus 2 x plus straight C end cell row blank equals cell 2 x to the power of 3 over 2 end exponent minus 2 over 5 x to the power of 5 over 2 end exponent plus 2 x plus straight C end cell row blank equals cell 2 x times x to the power of 1 half end exponent minus fraction numerator 2 x squared times x to the power of begin display style 1 half end style end exponent over denominator 5 end fraction plus 2 x plus straight C end cell row blank equals cell 2 x square root of x minus fraction numerator 2 x squared square root of x over denominator 5 end fraction plus 2 x plus straight C end cell end table

Dengan demikian, intergal tak tentu dari integral open parentheses 3 square root of x minus x square root of x plus 2 close parentheses d x adalah 2 x square root of x minus fraction numerator 2 x squared square root of x over denominator 5 end fraction plus 2 x plus straight C.

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

Hitunglah pengintegralan di bawah ini! 3)

Pembahasan Soal:

Integral fungsi undefined dapat ditentukan dengan rumus berikut. 

begin mathsize 14px style integral 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 end style 

sehingga hasil integral dari fungsi yang diberikan di atas dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral 4 x cubed minus 3 x squared plus 2 x minus 1 space straight d x end cell equals cell 4 over 4 x to the power of 4 minus 3 over 3 x cubed plus 2 over 2 x squared minus x plus C end cell row blank equals cell x to the power of 4 minus x cubed plus x squared minus x plus C end cell end table end style 

Dengan demikian, hasil integral fungsi yang diberikan adalah begin mathsize 14px style integral table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell x cubed end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus 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 cell x squared 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 2 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table 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 table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell x to the power of 4 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell x cubed 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 cell x squared end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus 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 plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table end style.

0

Roboguru

Dengan menjabarkan bentuk aljabar dari integral, temukan hasil integrasi di bawah ini.

Pembahasan Soal:

Rumus integral adalah sebagai berikut:

axndx=n+1axn+1+c 

Penyelesaiannya adalah sebagai berikut:

(1x3)2dx=====(1x3)(1x3)dx(12x3+x6)dxx3+12x3+1+6+11x6+1+cx42x4+71x7+cx21x4+71x7+c 

Jadi, hasil integrasi begin mathsize 14px style integral open parentheses 1 minus x cubed close parentheses squared space d x end style adalah x21x4+71x7+c.

0

Roboguru

Hitunglah pengintegralan di bawah ini! 2)

Pembahasan Soal:

Integral fungsi undefined dapat ditentukan dengan rumus berikut.

begin mathsize 14px style integral a x to the power of n space straight 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 end style 

sehingga dapat ditentukan hasil dari integral fungsi yang diberikan di atas sebagai berikut.

begin mathsize 14px style integral 7 x squared plus 3 space straight d x equals 7 over 3 x cubed plus 3 x plus C end style 

0

Roboguru

Tentukan integral dari .

Pembahasan Soal:

Gunakan konsep integral penjumlahan, pengurangan dan perkalian skalar dengan fungsi.


table attributes columnalign right center left columnspacing 2px end attributes row cell integral open square brackets f open parentheses x close parentheses plus g open parentheses x close parentheses close square brackets space straight d x end cell equals cell integral f open parentheses x close parentheses space straight d x plus integral g open parentheses x close parentheses space straight d x end cell row cell integral open square brackets f open parentheses x close parentheses minus g open parentheses x close parentheses close square brackets space straight d x end cell equals cell integral f open parentheses x close parentheses space straight d x minus integral g open parentheses x close parentheses space straight d x end cell row cell integral a x to the power of n 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 a space straight d x end cell equals cell a x plus c end cell end table


Akan ditentukan integral dari integral open parentheses 2 x squared minus 3 x plus 1 close parentheses space straight d x.

Perhatikan perhitungan berikut.


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


Jadi, diperoleh hasil integral dari integral open parentheses 2 x squared minus 3 x plus 1 close parentheses space straight d x adalah 2 over 3 x cubed minus 3 over 2 x squared plus x plus c.

0

Roboguru

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.

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Roboguru

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