Pertanyaan

Jika , maka mn = .....

Jika $begin mathsize 14px style integral fraction numerator x squared over denominator square root of m squared minus x squared end root end fraction d x equals 18 arc space sin invisible function application open parentheses n x close parentheses minus fraction numerator x square root of m squared minus x squared end root over denominator 2 end fraction plus C end style$ , maka mn = .....

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N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Pembahasan

Misalkan untuk . Maka Didapat pula dx = m cos⁡θ dθ sehingga Ingat bahwa , maka Diketahui bahwa ,perhatikan gambar di bawah ini. Didapat bahwa , sehingga Sehingga didapat bahwa Pada soal diketahui bahwa Maka dapat diperhatikan bahwa Sehingga

$begin mathsize 14px style integral fraction numerator x squared over denominator square root of m squared minus x squared end root end fraction d x end style$

Misalkan untuk .

Maka

Didapat pula
dx = m cos⁡θ dθ

sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator x squared over denominator square root of m squared minus x squared end root end fraction d x end cell equals cell integral fraction numerator open parentheses m sin invisible function application theta close parentheses squared over denominator square root of m squared minus open parentheses m sin invisible function application theta close parentheses squared end root end fraction open parentheses m cos invisible function application theta d theta close parentheses end cell row blank equals cell integral fraction numerator m squared sin squared invisible function application theta over denominator square root of m squared minus m squared sin squared invisible function application theta end root end fraction m cos invisible function application theta d theta end cell row blank equals cell integral fraction numerator m squared sin squared invisible function application theta over denominator square root of m squared open parentheses 1 minus sin squared invisible function application theta close parentheses end root end fraction m cos invisible function application theta d theta end cell row blank equals cell integral fraction numerator m squared sin squared invisible function application theta over denominator square root of m squared cos squared invisible function application theta end root end fraction m cos invisible function application theta d theta end cell row blank equals cell integral fraction numerator m squared sin squared invisible function application theta over denominator m cos invisible function application theta end fraction m cos invisible function application theta d theta end cell row blank equals cell integral m squared sin squared invisible function application theta d theta end cell row blank equals cell m squared integral sin squared invisible function application theta d theta end cell end table end style$

Ingat bahwa , maka

Diketahui bahwa , perhatikan gambar di bawah ini.

Didapat bahwa $begin mathsize 14px style cos invisible function application theta equals fraction numerator square root of m squared minus x squared end root over denominator m end fraction end style$ , sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half m squared open parentheses theta minus sin invisible function application theta cos invisible function application theta close parentheses plus C end cell equals cell open parentheses arcsin invisible function application x over m minus x over m fraction numerator square root of m squared minus x squared end root over denominator m end fraction close parentheses plus C end cell row blank equals cell 1 half m squared arcsin invisible function application x over m minus 1 half m squared x over m fraction numerator square root of m squared minus x squared end root over denominator m end fraction plus C end cell row blank equals cell 1 half m squared arcsin invisible function application x over m minus fraction numerator x square root of m squared minus x squared end root over denominator 2 end fraction plus C end cell end table end style$

Sehingga didapat bahwa

$begin mathsize 14px style integral fraction numerator x squared over denominator square root of m squared minus x squared end root end fraction d x equals 1 half m squared arcsin invisible function application x over m minus fraction numerator x square root of m squared minus x squared end root over denominator 2 end fraction plus C end style$

Pada soal diketahui bahwa

$begin mathsize 14px style integral fraction numerator x squared over denominator square root of m squared minus x squared end root end fraction d x equals 18 arcsin invisible function application open parentheses n x close parentheses minus fraction numerator x square root of m squared minus x squared end root over denominator 2 end fraction plus C end style$

Maka dapat diperhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half m squared end cell equals 18 row cell 1 over m end cell equals n end table end style

Sehingga