Pertanyaan

∫ a rc sin x d x = …

$\int a rc sin x d x = \dots$

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Pertama – tama cari terlebih dahulu turunan dari arc sin x Misalkan Maka Sekarang dengan menggunakan integral parsial

Pertama – tama cari terlebih dahulu turunan dari arc sin x

Misalkan

$begin mathsize 12px style fraction numerator d x over denominator d y end fraction equals cos space y end style$

Maka $begin mathsize 12px style fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator cos space y end fraction equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction end style$

Sekarang dengan menggunakan integral parsial

$begin mathsize 12px style u equals a r c sin space x d u equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x d v equals d x left right double arrow v equals x integral u d v equals u v minus integral v d u integral arcsin space x space d x equals x space arcsin space x minus integral open parentheses fraction numerator x over denominator square root of 1 minus x squared end root end fraction close parentheses d x integral arcsin space x space d x equals x space arcsin space x minus integral open parentheses fraction numerator x over denominator square root of 1 minus x squared end root end fraction close parentheses fraction numerator d open parentheses 1 minus x squared close parentheses over denominator negative 2 x end fraction integral arcsin space x space d x equals x space arcsin space x plus 1 half integral open parentheses 1 minus x squared close parentheses to the power of negative 1 half end exponent d open parentheses 1 minus x squared close parentheses integral arcsin space x space d x equals x space arcsin space x plus 1 half.2 square root of 1 minus x squared end root plus C integral arcsin space x space d x equals x space arcsin space x plus square root of 1 minus x squared end root plus C end style$