Pertanyaan

∫ a rc sin x d x = .....

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

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Jawaban terverifikasi

Pembahasan

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

Pertama – tama cari terlebih dahulu turunan dari arc sin x

Misalkan y = arcsin x

sin y = x

$fraction numerator d x over denominator d y end fraction space equals space cos space y$

Maka $fraction numerator d x over denominator d y end fraction space equals space fraction numerator 1 over denominator cos begin display style space end style begin display style y end style end fraction space equals space fraction numerator 1 over denominator square root of 1 space minus space x squared end root end fraction$
Sekarang dengan menggunakan integral parsial

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