Pertanyaan

∫ sin 4 x cos 4 x d x adalah ...

$\int sin^{4} x cos^{4} x d x$ adalah ...

$begin mathsize 12px style fraction numerator 3 x over denominator 128 end fraction minus 1 over 64 sin space 2 x plus 1 over 256 sin space 4 x plus C end style$
$begin mathsize 12px style fraction numerator 3 x over denominator 128 end fraction plus 1 over 64 sin space 2 x plus 1 over 256 sin space 4 x plus C end style$
$begin mathsize 12px style fraction numerator 3 x over denominator 64 end fraction minus 1 over 128 sin space 4 x plus 1 over 512 cos space 4 x plus C end style$
$begin mathsize 12px style fraction numerator 3 x over denominator 128 end fraction minus 32 space sin space 4 x plus 124 space sin space 8 x plus C end style$
$begin mathsize 12px style fraction numerator 3 x over denominator 128 end fraction minus 1 over 128 sin space 4 x plus 1 over 1024 sin space 8 x plus C end style$

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Klaim

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Karena kedua pangkatnya bilangan genap, untuk menentukan penyelesaiannya gunakan : Kesamaan setengah sudut dan Sehingga Jadi jawaban yang tepat adalah E.

Karena kedua pangkatnya bilangan genap, untuk menentukan penyelesaiannya gunakan :

Kesamaan setengah sudut $begin mathsize 12px style sin squared space x equals fraction numerator 1 minus cos space 2 x over denominator 2 end fraction end style$ dan $begin mathsize 12px style cos squared space x equals fraction numerator 1 plus cos space 2 x over denominator 2 end fraction end style$

Sehingga

$begin mathsize 12px style integral sin to the power of 4 x space cos to the power of 4 x space d x equals integral open parentheses sin squared x close parentheses squared open parentheses cos squared x close parentheses squared d x equals integral open parentheses fraction numerator 1 minus cos 2 x over denominator 2 end fraction close parentheses squared open parentheses fraction numerator 1 plus cos 2 x over denominator 2 end fraction close parentheses squared d x equals 1 over 16 integral left parenthesis 1 minus 2 space cos 2 x plus cos squared 2 x right parenthesis open parentheses 1 plus 2 space cos 2 x plus cos squared 2 x close parentheses d x equals 1 over 16 integral open parentheses 1 minus 2 cos squared 2 x plus cos to the power of 4 2 x close parentheses d x equals 1 over 16 integral d x minus 1 over 8 integral cos squared 2 space x space d x plus 1 over 16 integral cos to the power of 4 2 space x space d x equals 1 over 16 integral d x minus 1 over 8 integral fraction numerator 1 plus cos 4 x over denominator 2 end fraction plus 1 over 16 integral open parentheses fraction numerator 1 plus cos 4 x over denominator 2 end fraction close parentheses squared d x equals 1 over 16 integral d x minus 1 over 8 integral fraction numerator 1 plus cos 4 x over denominator 2 end fraction plus 1 over 64 integral open parentheses 1 plus 2 space cos 4 x plus cos squared 4 x close parentheses squared d x equals 1 over 16 integral d x minus 1 over 8 integral fraction numerator 1 plus cos 4 x over denominator 2 end fraction plus 1 over 64 integral d x plus 1 over 32 integral cos 4 x space d x plus 1 over 64 integral open parentheses fraction numerator 1 plus cos 8 x over denominator 2 end fraction close parentheses d x equals 1 over 16 integral d x minus 1 over 8 integral fraction numerator 1 plus cos 4 x over denominator 2 end fraction plus 1 over 64 integral d x plus 1 over 32 integral cos 4 x space d x plus 1 over 128 integral d x plus 1 over 128 integral cos 8 x space d x equals 1 over 16 integral d x minus 1 over 16 integral d x minus 1 over 16 integral cos 4 x space d x plus 1 over 64 integral d x plus 1 over 32 integral cos 4 x space d x plus 1 over 128 integral d x plus integral 1 over 128 cos 8 x space d x equals 3 over 128 integral d x minus 1 over 32 integral cos 4 x space d x plus 1 over 128 integral cos 8 x space d x equals fraction numerator 3 x over denominator 128 end fraction minus 1 over 128 sin 4 x plus 1 over 1024 sin 8 x plus C end style$

Jadi jawaban yang tepat adalah E.