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Nilai dari ∫02π​​sin 5x⋅sin x dx=...

Pertanyaan

Nilai dari integral subscript 0 superscript straight pi over 2 end superscript sin space 5 x times sin space x space d x equals... 

  1. negative 1 over 6 

  2. negative 1 over 12 

  3. 0 

  4. 1 over 8 

  5. 5 over 12 

A. Armanda

Master Teacher

Mahasiswa/Alumni Universitas Indraprasta PGRI

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript a superscript b f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses space d x end cell equals cell integral subscript a superscript b f open parentheses x close parentheses space d x plus-or-minus integral subscript a superscript b g open parentheses x close parentheses space d x end cell row cell integral subscript a superscript b k f open parentheses x close parentheses space d x end cell equals cell k integral subscript a superscript b f open parentheses x close parentheses space d x end cell row cell integral subscript a superscript b cos space x space d x end cell equals cell open square brackets sin space x close square brackets subscript a superscript b end cell end table 

dengan k adalah konstanta.

Karena

sin space alpha times sin space beta equals negative 1 half open square brackets cos space open parentheses alpha plus beta close parentheses minus cos space open parentheses alpha minus beta close parentheses close square brackets 

maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 5 x times sin space x end cell equals cell negative 1 half open square brackets cos space open parentheses 5 x plus x close parentheses minus cos space open parentheses 5 x minus x close parentheses close square brackets end cell row blank equals cell negative 1 half open parentheses cos space 6 x minus cos space 4 x close parentheses end cell end table 

Sehingga, 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral subscript 0 superscript straight pi over 2 end superscript sin space 5 x times sin space x space d x end cell row blank equals cell integral subscript 0 superscript straight pi over 2 end superscript minus 1 half open parentheses cos space 6 x minus cos space 4 x close parentheses space d x end cell row blank equals cell negative 1 half integral subscript 0 superscript straight pi over 2 end superscript open parentheses cos space 6 x minus cos space 4 x close parentheses space d x end cell row blank equals cell negative 1 half open square brackets integral subscript 0 superscript straight pi over 2 end superscript cos space 6 x space d x minus integral subscript 0 superscript straight pi over 2 end superscript cos space 4 x space d x close square brackets end cell row blank equals cell negative 1 half open parentheses open square brackets sin space 6 x close square brackets subscript 0 superscript straight pi over 2 end superscript minus open square brackets sin space 4 x close square brackets subscript 0 superscript straight pi over 2 end superscript close parentheses end cell row blank equals cell negative 1 half open square brackets open parentheses sin space 6 times straight pi over 2 minus sin space 6 times 0 close parentheses minus open parentheses sin space 4 times straight pi over 2 minus sin space 4 times 0 close parentheses close square brackets end cell row blank equals cell negative 1 half open square brackets sin space open parentheses 1 times 360 degree plus 180 degree close parentheses minus 0 minus open parentheses sin space 360 degree minus 0 close parentheses close square brackets end cell row blank equals cell negative 1 half open parentheses sin space 180 degree minus sin space 360 degree close parentheses end cell row blank equals cell negative 1 half open parentheses 0 minus 0 close parentheses end cell row blank equals cell negative 1 half times 0 end cell row blank equals 0 end table 

Jadi, jawaban yang tepat adalah C.

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