Pertanyaan

Jika hasil dari dapat dinyatakan dalam bentuk , maka nilai dari P + Q + R = ….

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Pembahasan

Misalkan dan Didapat bahwa dan Maka, Kemudian dengan cara yang sama kita misalkan : dan Didapat bahwa Dan Jadi, Akibatnya, Maka diperoleh P = 9, Q = -6, dan R = 2. Jadi,

Misalkan

dan

$begin mathsize 14px style fraction numerator d v over denominator d x end fraction equals e to the power of 3 x end exponent end style$

Didapat bahwa

$begin mathsize 14px style fraction numerator d u over denominator d x end fraction equals 2 x end style$

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row v equals cell integral fraction numerator d v over denominator d x end fraction blank d x end cell row blank equals cell integral e to the power of 3 x end exponent blank d x end cell row blank equals cell 1 third e to the power of 3 x end exponent plus C end cell end table end style$

Maka,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral open parentheses x squared e to the power of 3 x end exponent close parentheses blank d x end cell equals cell integral u times fraction numerator d v over denominator d x end fraction d x end cell row blank equals cell u times v minus integral v times fraction numerator d u over denominator d x end fraction d x end cell row blank equals cell x squared times 1 third e to the power of 3 x end exponent minus integral open parentheses blank 1 third e to the power of 3 x end exponent times 2 x close parentheses d x end cell row blank equals cell 1 third x squared e to the power of 3 x end exponent minus 2 over 3 integral x e to the power of 3 x end exponent d x plus C end cell end table end style$

Kemudian dengan cara yang sama kita misalkan :

dan

$begin mathsize 14px style fraction numerator d q over denominator d x end fraction equals e to the power of 3 x end exponent end style$

Didapat bahwa

$begin mathsize 14px style fraction numerator d p over denominator d x end fraction equals 1 end style$

Dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row q equals cell integral fraction numerator d q over denominator d x end fraction blank d x end cell row blank equals cell integral e to the power of 3 x end exponent blank d x end cell row blank equals cell 1 third e to the power of 3 x end exponent plus C end cell end table end style$

Jadi,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral x e to the power of 3 x end exponent d x end cell equals cell integral p times fraction numerator d q over denominator d x end fraction d x end cell row blank equals cell p times q minus integral q times fraction numerator d p over denominator d x end fraction d x end cell row blank equals cell x times 1 third e to the power of 3 x end exponent minus integral 1 third e to the power of 3 x end exponent times 1 d x end cell row blank equals cell 1 third x e to the power of 3 x end exponent minus 1 over 9 e to the power of 3 x end exponent plus C end cell end table end style$

Akibatnya,

Maka diperoleh P = 9, Q = -6, dan R = 2.

Jadi,