Pertanyaan

Hasil dari ∫ x 2 1 + x 2 1 d x = ....

Hasil dari $\int \frac{1}{x ^{2} 1 + x ^{2}} d x =$ ....

$begin mathsize 14px style fraction numerator square root of 1 plus x squared end root over denominator x end fraction plus C end style$
$begin mathsize 14px style fraction numerator x over denominator square root of 1 plus x squared end root end fraction plus C end style$
$begin mathsize 14px style negative fraction numerator x over denominator square root of 1 plus x squared end root end fraction plus C end style$
$begin mathsize 14px style negative fraction numerator square root of 1 plus x squared end root over denominator x end fraction plus C end style$

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Misalkan: Sehingga diperoleh y adalah sisi miring sebuah segitiga siku-siku dengan kedua sisi tegaknya 1 dan . Maka diperoleh dan Jika kita integralkan kedua ruas maka diperoleh Sehingga, Misalkan: Sehingga, Maka integral di atas menjadi Jadi, jawaban yang tepat adalah D.

Misalkan:

Sehingga diperoleh y adalah sisi miring sebuah segitiga siku-siku dengan kedua sisi tegaknya 1 dan .

Maka diperoleh

$begin mathsize 14px style cosec invisible function application theta equals fraction numerator square root of 1 plus x squared end root over denominator x end fraction end style$

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan invisible function application theta end cell equals cell x over 1 end cell row x equals cell tan invisible function application theta end cell row cell fraction numerator d x over denominator d theta end fraction end cell equals cell sec squared invisible function application theta end cell end table end style$

Jika kita integralkan kedua ruas maka diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator d x over denominator d theta end fraction d theta end cell equals cell integral sec squared invisible function application theta d theta end cell row cell integral d x end cell equals cell integral sec squared invisible function application theta d theta end cell end table end style$

Sehingga,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 1 over denominator x squared square root of 1 plus x squared end root end fraction blank d x end cell equals cell integral fraction numerator 1 over denominator tan squared invisible function application theta square root of 1 plus tan squared invisible function application theta end root end fraction times sec squared invisible function application theta d theta end cell row blank equals cell integral fraction numerator 1 over denominator tan squared invisible function application theta square root of 1 plus tan squared invisible function application theta end root end fraction times sec squared invisible function application theta d theta end cell row blank equals cell integral fraction numerator 1 over denominator tan squared invisible function application theta square root of sec squared invisible function application theta end root end fraction times sec squared invisible function application theta d theta end cell row blank equals cell integral fraction numerator 1 over denominator tan squared invisible function application theta invisible function application sec invisible function application theta end fraction times sec squared invisible function application theta d theta end cell row blank equals cell integral fraction numerator sec invisible function application theta over denominator tan squared invisible function application theta end fraction d theta end cell row blank equals cell integral fraction numerator 1 over denominator cos invisible function application theta end fraction times fraction numerator cos squared invisible function application theta over denominator sin squared invisible function application theta end fraction d theta end cell row blank equals cell integral fraction numerator cos invisible function application theta over denominator sin squared invisible function application theta end fraction d theta end cell end table end style$

Misalkan:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell sin invisible function application theta end cell row cell fraction numerator d u over denominator d theta end fraction end cell equals cell cos invisible function application theta end cell row cell integral fraction numerator d u over denominator d theta end fraction d theta end cell equals cell integral cos invisible function application theta d theta end cell row cell integral d u end cell equals cell integral cos invisible function application theta d theta end cell end table end style$

Sehingga,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator cos invisible function application theta over denominator sin squared invisible function application theta end fraction d theta end cell equals cell integral 1 over u squared d u end cell row blank equals cell integral u to the power of negative 2 end exponent blank d u end cell row blank equals cell negative 1 over u plus C end cell row blank equals cell negative fraction numerator 1 over denominator sin invisible function application theta end fraction plus C end cell row blank equals cell negative csc invisible function application theta blank end cell end table end style$

Maka integral di atas menjadi

Jadi, jawaban yang tepat adalah D.