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Hasil dari ∫(1+sin x1− sin x​)dx= ...

Pertanyaan

Hasil dari integral open parentheses fraction numerator 1 minus space sin space x over denominator 1 plus sin space x end fraction close parentheses d x equals ...

  1. 2 space tan space x minus 2 space sec space x minus x plus straight C

  2. 2 space cot space x minus 2 space sec space x minus x plus straight C

  3. 2 space tan space x minus sec space x plus x plus straight C

  4. negative 2 space tan space x plus 2 space sec space x minus x plus straight C

  5. 2 space cot space x minus 2 space csc space x minus x plus straight C

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Ingatlah:

Rumus Identitas Trigonometri, yaitu:

table attributes columnalign center center left end attributes row cell fraction numerator 1 over denominator cos squared space x end fraction end cell equals cell sec squared space x end cell row cell fraction numerator sin squared italic space x over denominator cos squared italic space x end fraction end cell equals cell tan squared italic space x end cell row cell tan squared space x end cell equals cell sec squared space x minus 1 end cell end table

Rumus Intergral Fungsi Trigonometri, yaitu:

table attributes columnalign left center center end attributes row cell integral sec squared space x space d x end cell equals cell tan space x plus straight C end cell row cell integral sec space x times tan space x space d x end cell equals cell sec space x plus straight C end cell end table

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral open parentheses fraction numerator 1 minus sin space x over denominator 1 plus sin space x end fraction close parentheses d x end cell row blank equals cell integral open parentheses fraction numerator 1 minus sin space x over denominator 1 plus sin space x end fraction times fraction numerator 1 minus sin space x over denominator 1 minus sin space x end fraction close parentheses d x end cell row blank equals cell integral fraction numerator open parentheses 1 minus sin space x close parentheses squared over denominator 1 minus sin squared space x end fraction d x end cell row blank equals cell integral fraction numerator 1 minus 2 space sin space x plus sin squared space x over denominator cos squared space x end fraction d x end cell row blank equals cell integral open parentheses fraction numerator 1 over denominator cos squared space x end fraction minus fraction numerator 2 over denominator cos squared space x end fraction times fraction numerator sin space x over denominator cos squared space x end fraction plus fraction numerator sin squared space x over denominator cos squared space x end fraction close parentheses d x end cell row blank equals cell integral sec squared space x space d x minus 2 integral sec space x times tan space x space d x plus integral tan squared space x space d x end cell row blank equals cell integral sec squared space x space d x minus 2 integral sec space x times tan space x space d x plus integral left parenthesis sec squared space x minus 1 right parenthesis space d x end cell row blank equals cell integral sec squared space x space d x minus 2 integral sec space x times tan space x plus integral sec squared space x space d x minus integral 1 space d x end cell row blank equals cell tan space x minus 2 space sec space x plus tan space x minus x plus straight C end cell row blank equals cell tan italic space x plus tan space x italic minus 2 italic space sec italic space x italic minus x plus straight C end cell row blank equals cell 2 space tan space x minus 2 space sec space x minus x plus straight C end cell end table

Hasil dari integral open parentheses fraction numerator 1 minus space sin space x over denominator 1 plus sin space x end fraction close parentheses d x equals 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 space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank tan 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 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 2 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 sec 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 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 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 straight C end table.

Jadi, jawaban yang tepat adalah A.

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