Pertanyaan

Hitung. 4 . ∫ 0 2 1 1 − x 2 sin − 1 x d x

Hitung.

$4$ . $\int_{0}^{\frac{1}{2}} \frac{sin ^{- 1} x}{1 - x ^{2}} d x$

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Misal: , maka: Diperoleh integralnya yaitu: Maka, hasil integral dari adalah .

Misal: t equals sin to the power of negative 1 end exponent space x , maka:

$table attributes columnalign right center left columnspacing 0px end attributes row t equals cell sin to the power of negative 1 end exponent space x end cell row cell d t end cell equals cell fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x end cell row cell d x end cell equals cell square root of 1 minus x squared end root space d t end cell end table$

Diperoleh integralnya yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 0 superscript 1 half end superscript fraction numerator sin to the power of negative 1 end exponent space x over denominator square root of 1 minus x squared end root end fraction d x end cell equals cell integral subscript 0 superscript 1 half end superscript sin to the power of negative 1 end exponent space x times fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction d x end cell row blank equals cell integral subscript 0 superscript 1 half end superscript space t space d t end cell row blank equals cell right enclose t squared over 2 end enclose subscript space 0 end subscript superscript space 1 half end superscript end cell row blank equals cell right enclose open parentheses sin to the power of negative 1 end exponent space x close parentheses squared over 2 end enclose subscript space 0 end subscript superscript space 1 half end superscript end cell row blank equals cell fraction numerator sin to the power of negative 1 end exponent open parentheses begin display style 1 half end style close parentheses squared over denominator 2 end fraction minus fraction numerator sin to the power of negative 1 end exponent open parentheses 0 close parentheses squared over denominator 2 end fraction end cell row blank equals cell fraction numerator begin display style straight pi squared over 36 end style over denominator 2 end fraction minus 0 end cell row blank equals cell straight pi squared over 36 times 1 half end cell row blank equals cell straight pi squared over 72 end cell end table$

Maka, hasil integral dari $integral subscript 0 superscript 1 half end superscript fraction numerator sin to the power of negative 1 end exponent space x over denominator square root of 1 minus x squared end root end fraction d x$ adalah straight pi squared over 72 .