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Pertanyaan

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis end style adalah fungsi genap. Jika begin mathsize 14px style integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis d x equals 14 end style dan begin mathsize 14px style integral subscript 1 superscript 3 f left parenthesis x right parenthesis d x equals negative 2 end style, maka begin mathsize 14px style integral subscript negative 1 end subscript superscript 1 f left parenthesis x right parenthesis d x equals end style ....

  1. 18 

  2. 15 begin mathsize 14px style space end style

  3. 12 begin mathsize 14px style space end style

  4. 10 begin mathsize 14px style space end style

  5. begin mathsize 14px style space end style

Pembahasan Video:

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis d x end cell equals cell integral subscript negative 3 end subscript superscript negative 1 end superscript f left parenthesis x right parenthesis d x plus integral subscript negative 1 end subscript superscript 1 f left parenthesis x right parenthesis d x plus integral subscript 1 superscript 3 f left parenthesis x right parenthesis d x end cell row cell integral subscript negative 1 end subscript superscript 1 f left parenthesis x right parenthesis d x end cell equals cell integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis d x minus integral subscript negative 3 end subscript superscript negative 1 end superscript f left parenthesis x right parenthesis d x minus integral subscript 1 superscript 3 f left parenthesis x right parenthesis d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis d x minus open parentheses 2 times integral subscript 1 superscript 3 f left parenthesis x right parenthesis d x close parentheses space left parenthesis ingat space fungsi space genap right parenthesis end cell row blank equals cell 14 minus left parenthesis negative 4 right parenthesis end cell row blank equals 18 end table end style 

 

Jadi, jawaban yang tepat adalah A.

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Terakhir diupdate 03 Mei 2021

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Fungsi  memenuhi . Jika nilai  dan , maka nilai

Pembahasan Soal:

Diketahui fungsi f left parenthesis x right parenthesis memenuhi f left parenthesis x right parenthesis equals f left parenthesis negative x right parenthesis merupakan fungsi genap yang memenuhi sifat integral subscript negative a end subscript superscript a f left parenthesis x right parenthesis space d x equals 2 integral subscript 0 superscript a f left parenthesis x right parenthesis space d x.

Secara umum, bentuk integral juga dapat diubah batasnya seperti berikut, integral subscript a superscript c f left parenthesis x right parenthesis space d x equals integral subscript a superscript b f left parenthesis x right parenthesis space d x plus integral subscript b superscript c f left parenthesis x right parenthesis space d x untuk rentang a less or equal than b less or equal than c.

Diberikan nilai integral integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis space d x equals 6, maka bentuknya dapat diubah menjadi 

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 3 end subscript superscript 3 f left parenthesis x right parenthesis space d x end cell equals 6 row cell 2 integral subscript 0 superscript 3 f left parenthesis x right parenthesis space d x end cell equals 6 row cell integral subscript 0 superscript 3 f left parenthesis x right parenthesis space d x end cell equals cell 6 over 2 end cell row cell integral subscript 0 superscript 3 f left parenthesis x right parenthesis space d x end cell equals 3 end table 

Diketahui pula nilai  integral subscript 2 superscript 3 f left parenthesis x right parenthesis space d x equals 1, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 0 superscript 3 f left parenthesis x right parenthesis space d x end cell equals cell integral subscript 0 superscript 2 f left parenthesis x right parenthesis space d x plus integral subscript 2 superscript 3 f left parenthesis x right parenthesis space d x end cell row cell integral subscript 0 superscript 2 f left parenthesis x right parenthesis space d x end cell equals cell integral subscript 0 superscript 3 f left parenthesis x right parenthesis space d x minus integral subscript 2 superscript 3 f left parenthesis x right parenthesis space d x end cell row blank equals cell 3 minus 1 end cell row blank equals 2 end table 

Jadi, nilai dari integral subscript 0 superscript 2 f left parenthesis x right parenthesis space d x equals 2.

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Diketahui  adalah fungsi genap. Jika  maka

Pembahasan Soal:

Diketahui begin mathsize 14px style f open parentheses x close parentheses end style adalah fungsi genap, berlaku begin mathsize 14px style integral subscript negative a end subscript superscript a f left parenthesis x right parenthesis d x equals 2 integral subscript 0 superscript a f left parenthesis x right parenthesis d x end style

Ingat: begin mathsize 14px style integral subscript a superscript c f left parenthesis x right parenthesis space d x equals integral subscript a superscript b f left parenthesis x right parenthesis space d x space plus integral subscript b superscript c f left parenthesis x right parenthesis space d x space dengan space a less than b less than c end style

Sehingga

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 5 end subscript superscript 5 f left parenthesis x right parenthesis d x end cell equals 14 row cell 2 integral subscript 0 superscript 5 f left parenthesis x right parenthesis d x end cell equals 14 row cell integral subscript 0 superscript 5 f left parenthesis x right parenthesis d x end cell equals 7 row cell integral subscript 0 superscript 1 f left parenthesis x right parenthesis d x plus integral subscript 1 superscript 5 f left parenthesis x right parenthesis d x end cell equals 7 row cell integral subscript 0 superscript 1 f left parenthesis x right parenthesis d x plus 10 end cell equals 7 row cell integral subscript 0 superscript 1 f left parenthesis x right parenthesis d x end cell equals cell negative 3 end cell end table end style

Diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 1 end subscript superscript 1 f left parenthesis x right parenthesis d x end cell equals cell 2 integral subscript 0 superscript 1 f left parenthesis x right parenthesis d x end cell row blank equals cell 2 times open parentheses negative 3 close parentheses end cell row blank equals cell negative 6 end cell end table end style

Jadi, jawaban yang tepat adalah B.
 

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Nilai dari

Pembahasan Soal:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell integral subscript negative 1 end subscript superscript 1 open parentheses 1 third x squared plus 3 x minus 2 close parentheses d x end cell row blank equals cell open square brackets 1 third times fraction numerator 1 over denominator 2 plus 1 end fraction x to the power of 2 plus 1 end exponent plus fraction numerator 3 over denominator 1 plus 1 end fraction x to the power of 1 plus 1 end exponent minus fraction numerator 2 over denominator 0 plus 1 end fraction x to the power of 0 plus 1 end exponent close square brackets subscript negative 1 end subscript superscript 1 end cell row blank equals cell open square brackets 1 over 9 x cubed plus 3 over 2 x squared minus 2 x close square brackets subscript negative 1 end subscript superscript 1 end cell row blank equals cell open parentheses 1 over 9 open parentheses 1 close parentheses cubed plus 3 over 2 open parentheses 1 close parentheses squared minus 2 open parentheses 1 close parentheses close parentheses minus end cell row blank blank cell open parentheses 1 over 9 open parentheses negative 1 close parentheses cubed plus 3 over 2 open parentheses negative 1 close parentheses squared minus 2 open parentheses negative 1 close parentheses close parentheses end cell row blank equals cell open parentheses 1 over 9 plus up diagonal strike 3 over 2 end strike minus 2 close parentheses minus open parentheses negative 1 over 9 plus up diagonal strike 3 over 2 end strike plus 2 close parentheses end cell row blank equals cell 2 over 9 minus 4 end cell row blank equals cell fraction numerator 2 minus 36 over denominator 9 end fraction end cell row blank equals cell fraction numerator negative 34 over denominator 9 end fraction end cell row blank equals cell negative 34 over 9 end cell end table end style     

Jadi, jawaban yang tepat adalah A.

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Pembahasan Soal:

integral subscript negative 2 end subscript superscript 2 left parenthesis 3 x squared space plus space 4 right parenthesis d x space equals space open square brackets 1 third. space 3 x cubed space plus space 4. x close square brackets subscript negative 2 end subscript superscript 2  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 space open square brackets x cubed space plus space 4 x close square brackets subscript negative 2 end subscript superscript 2  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 space open square brackets left parenthesis 2 right parenthesis cubed space plus space 4 left parenthesis 2 right parenthesis close square brackets space minus space open square brackets left parenthesis negative 2 right parenthesis cubed space plus space 4 left parenthesis negative 2 right parenthesis close square brackets  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 space 16 space plus space 16  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 space 32

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Jika  dengan  merupakan fungsi genap dan  merupakan fungsi ganjil, maka hasil dari adalah ...

Pembahasan Soal:

Perhatikan perhitungan berikut!

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 3 end subscript superscript 3 f open parentheses x close parentheses open parentheses g open parentheses x close parentheses minus 1 close parentheses space d x end cell equals cell integral subscript negative 3 end subscript superscript 3 open square brackets f open parentheses x close parentheses g open parentheses x close parentheses minus f open parentheses x close parentheses close square brackets space d x end cell row 6 equals cell integral subscript negative 3 end subscript superscript 3 f open parentheses x close parentheses g open parentheses x close parentheses space d x minus integral subscript negative 3 end subscript superscript 3 f open parentheses x close parentheses space d x end cell end table end style

Diketajui undefined merupakan fungsi genap dan undefined merupakan fungsi ganjil. Akibatnya, didapat hubungan berikut ini.

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses negative x close parentheses end cell equals cell f open parentheses x close parentheses end cell row cell g open parentheses negative x close parentheses end cell equals cell negative g open parentheses x close parentheses end cell end table

Misal begin mathsize 14px style h open parentheses x close parentheses equals f open parentheses x close parentheses g open parentheses x close parentheses end style, maka didapat hasil perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell h open parentheses negative x close parentheses end cell equals cell f open parentheses negative x close parentheses g open parentheses negative x close parentheses end cell row blank equals cell negative f open parentheses x close parentheses g open parentheses x close parentheses end cell row blank equals cell negative h open parentheses x close parentheses end cell end table end style

Karena h open parentheses negative x close parentheses equals negative h open parentheses x close parentheses,maka  undefined merupakan fungsi ganjil.

Oleh karena itu, didapat perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row 6 equals cell integral subscript negative 3 end subscript superscript 3 f open parentheses x close parentheses g open parentheses x close parentheses space d x minus integral subscript negative 3 end subscript superscript 3 f open parentheses x close parentheses space d x end cell row 6 equals cell 0 minus 2 integral subscript 0 superscript 3 f open parentheses x close parentheses space d x end cell row 6 equals cell negative 2 integral subscript 0 superscript 3 f open parentheses x close parentheses space d x end cell row cell negative 3 end cell equals cell integral subscript 0 superscript 3 f open parentheses x close parentheses space d x end cell end table

Dengan demikian, hasil dariintegral subscript 0 superscript 3 f open parentheses x close parentheses space d x adalah negative 3.

Oleh karena itu, jawaban yang tepat adalah B.

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