Pertanyaan

Hasil dari ∫ 4 x − x 2 d x adalah...

Hasil dari $\int 4 x - x^{2} d x$ adalah...

$2 space a r c space sin open parentheses fraction numerator 2 plus x over denominator 2 end fraction close parentheses plus open parentheses fraction numerator 2 plus x over denominator 2 end fraction close parentheses square root of 4 x plus x squared end root plus C$
$2 space a r c space sin open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses minus open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses square root of 4 x minus x squared end root plus C$
$2 space a r c space sin open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses minus open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses square root of 4 x plus x squared end root plus C$

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

misalkan: perhatikan segitiga di bawah ini: karena sehingga diperoleh

misalkan:

$x equals 2 space sin space u space plus 2 space rightwards double arrow u space equals space a r c space sin space open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses d x equals 2 space cos space u space d u$

perhatikan segitiga di bawah ini:

karena $space sin space u space equals open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses$ sehingga diperoleh

$cos space u space equals fraction numerator s a m p i n g over denominator m i r i n g end fraction equals fraction numerator square root of 2 squared minus left parenthesis x minus 2 right parenthesis squared end root over denominator 2 end fraction equals fraction numerator square root of 4 minus x squared plus 4 x minus 4 end root over denominator 2 end fraction equals fraction numerator square root of 4 x minus x squared end root over denominator 2 end fraction sin space 2 u space equals space 2 space sin space u space cos space u space equals space 2 open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of 4 x minus x squared end root over denominator 2 end fraction close parentheses equals fraction numerator open parentheses x minus 2 close parentheses square root of 4 x minus x squared end root over denominator 2 end fraction m a k a comma integral subscript blank square root of 4 x minus x squared end root d x equals integral square root of x left parenthesis 4 minus x right parenthesis end root d x equals integral square root of left parenthesis 2 space sin space u plus 2 right parenthesis left parenthesis 4 space minus space 2 space sin u minus 2 right parenthesis end root left parenthesis 2 space cos space u space d u right parenthesis equals integral square root of open parentheses 2 plus 2 space sin space u close parentheses open parentheses 2 minus 2 space s i space u close parentheses end root left parenthesis 2 space cos space u space d u right parenthesis equals integral square root of open parentheses 4 minus 4 space sin squared u close parentheses end root left parenthesis 2 space cos space u space d u right parenthesis equals integral 2 square root of open parentheses 1 minus space sin squared u close parentheses end root open parentheses 2 space cos space u space d u close parentheses equals 2 integral subscript blank square root of cos squared u end root left parenthesis 2 space cos space u space d u right parenthesis equals 4 integral subscript blank cos space squared u space d u equals 4 integral fraction numerator cos space 2 space u plus 1 over denominator 2 end fraction d u equals 4 over 2 open parentheses 1 half sin space 2 u plus u close parentheses plus C equals 4 over 2 open parentheses 1 half fraction numerator open parentheses x minus 2 close parentheses square root of 4 x minus x squared end root over denominator 2 end fraction a r c space sin open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses close parentheses plus C equals fraction numerator open parentheses x minus 2 close parentheses over denominator 2 end fraction square root of 4 x minus x squared end root plus 2 space a r c space sin space open parentheses fraction numerator x minus 2 over denominator 2 end fraction close parentheses plus C$