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Hasil dari ∫secxdx=...

Pertanyaan

Hasil dari begin mathsize 14px style integral sec invisible function application x d x equals... end style

  1. begin mathsize 14px style ln invisible function application open vertical bar csc invisible function application x plus cot invisible function application x close vertical bar plus C end style

  2. begin mathsize 14px style ln invisible function application open vertical bar csc invisible function application x plus sec invisible function application x close vertical bar plus C end style

  3. begin mathsize 14px style ln invisible function application open vertical bar sec invisible function application x plus tan invisible function application x close vertical bar plus C end style

  4. begin mathsize 14px style ln invisible function application open vertical bar cos invisible function application x plus sin invisible function application x close vertical bar plus C end style

  5. begin mathsize 14px style ln invisible function application open vertical bar cos invisible function application x minus sin invisible function application x close vertical bar plus C end style

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Agar dapat menggunakan integral subsitusi perlu kita kalikan dengan

begin mathsize 14px style fraction numerator sec invisible function application x plus tan invisible function application x over denominator sec invisible function application x plus tan invisible function application x end fraction end style 

Sehingga fungsi tersebut menjadi,

begin mathsize 14px style integral sec invisible function application x d x equals integral open parentheses sec invisible function application x times fraction numerator sec invisible function application x plus tan invisible function application x over denominator sec invisible function application x plus tan invisible function application x end fraction close parentheses d x space space space space space space space space space space space space space space space space space space space space equals integral open parentheses fraction numerator sec squared invisible function application x plus sec invisible function application x tan invisible function application x over denominator sec invisible function application x plus tan invisible function application x end fraction close parentheses d x end style 

Kemudian kita gunakan integral substitusi dengan memisalkan u=secx+tanx  sehingga diperoleh,

begin mathsize 14px style fraction numerator d u over denominator d x end fraction equals sec invisible function application x plus tan invisible function application x d x equals fraction numerator d u over denominator sec invisible function application x tan invisible function application x plus sec squared invisible function application x end fraction end style

 

Maka,

begin mathsize 14px style integral sec invisible function application x d x equals integral open parentheses sec invisible function application x times fraction numerator sec invisible function application x plus tan invisible function application x over denominator sec invisible function application x plus tan invisible function application x end fraction close parentheses d x space space space space space space space space space space space space space space space space space space space space equals integral open parentheses fraction numerator sec squared invisible function application x plus sec invisible function application x tan invisible function application x over denominator sec invisible function application x plus tan invisible function application x end fraction close parentheses d x space space space space space space space space space space space space space space space space space space space space equals integral open parentheses fraction numerator sec squared invisible function application x plus sec invisible function application x tan invisible function application x over denominator u end fraction close parentheses times fraction numerator d u over denominator sec invisible function application x tan invisible function application x plus sec squared invisible function application x end fraction space space space space space space space space space space space space space space space space space space space space equals integral 1 over u d u space space space space space space space space space space space space space space space space space space space space equals ln invisible function application u plus C space space space space space space space space space space space space space space space space space space space space equals ln invisible function application open vertical bar sec invisible function application x plus tan invisible function application x close vertical bar plus C end style

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