Pertanyaan

Jika dan F(0)= 5, maka nilai dari adalah ....

Jika dan F(0) = 5, maka nilai dari adalah ....

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Pembahasan

Misalkan: Maka, Karena diketahui pula bahwa F(0) = 5 maka diperoleh nilai konstanta C sebagai berikut. Kemudian hitung nilai p dengan menggunakan fungsi F(x) di atas. Akibatnya, Sehingga diperoleh,

Misalkan:

Maka,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of apostrophe open parentheses x close parentheses end cell equals cell negative 3 sin invisible function application x plus 2 over pi integral from 0 to pi of f open parentheses x close parentheses d x end cell row cell f to the power of apostrophe open parentheses x close parentheses end cell equals cell negative 3 sin invisible function application x plus fraction numerator 2 p over denominator pi end fraction end cell row cell integral f to the power of apostrophe open parentheses x close parentheses d x end cell equals cell integral open parentheses negative 3 sin invisible function application x plus fraction numerator 2 p over denominator pi end fraction close parentheses d x end cell row cell straight F open parentheses x close parentheses end cell equals cell 3 cos invisible function application x plus fraction numerator 2 p x over denominator pi end fraction plus C end cell end table end style$

Karena diketahui pula bahwa F(0) = 5 maka diperoleh nilai konstanta C sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight F open parentheses 0 close parentheses end cell equals cell 3 cos invisible function application 0 plus fraction numerator 2 p open parentheses 0 close parentheses over denominator pi end fraction plus C end cell row 5 equals cell 3 open parentheses 1 close parentheses plus C end cell row 2 equals C end table end style$

Kemudian hitung nilai p dengan menggunakan fungsi F(x) di atas.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row p equals cell integral from 0 to pi of f open parentheses x close parentheses d x end cell row p equals cell open straight F open parentheses x close parentheses close square brackets subscript 0 superscript pi end cell row p equals cell straight F open parentheses pi close parentheses minus straight F open parentheses 0 close parentheses end cell row p equals cell 3 cos invisible function application pi plus fraction numerator 2 p times pi over denominator pi end fraction minus open square brackets 3 cos invisible function application 0 plus fraction numerator 2 p open parentheses 0 close parentheses over denominator pi end fraction close square brackets end cell row p equals cell 3 open parentheses negative 1 close parentheses plus 2 p minus 3 open parentheses 1 close parentheses minus 0 end cell row p equals cell negative 3 plus 2 p minus 3 end cell row p equals 6 end table end style$

Akibatnya,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight F open parentheses x close parentheses end cell equals cell 3 cos invisible function application x plus fraction numerator 2 p x over denominator pi end fraction plus C end cell row blank equals cell 3 cos invisible function application x plus fraction numerator 2 open parentheses 6 close parentheses x over denominator pi end fraction plus 2 end cell row blank equals cell 3 cos invisible function application x plus fraction numerator 12 x over denominator pi end fraction plus 2 end cell end table end style$

Sehingga diperoleh,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight F open parentheses pi over 2 close parentheses end cell equals cell 3 cos invisible function application pi over 2 plus fraction numerator 12 open parentheses pi over 2 close parentheses over denominator pi end fraction plus 2 end cell row blank equals cell 3 times 0 plus 6 plus 2 end cell row blank equals 8 end table end style$