Pertanyaan

Hasil dari ∫ x 2 e 2 x d x adalah ...

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

N. Rahayu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jakarta

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

Pembahasan

Dari soal diketahui Kita terapkan metode integral parsial pada bentuk integral di atas. Misalkan dan , maka dan Kemudian, perhatikan perhitungan berikut, Selanjutnya, perhatikan integral berikut. Akan digunakan metode integral parsial untuk menyelesaikan integral tersebut. Misalkan p = x dan , maka dan Sehingga diperoleh Jadi, jawaban yang tepat adalah B.

Dari soal diketahui

Kita terapkan metode integral parsial pada bentuk integral di atas.

Misalkan dan $begin mathsize 14px style fraction numerator d v over denominator d x end fraction equals e to the power of 2 x end exponent end style$ , maka

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

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row v equals cell integral fraction numerator d v over denominator d x end fraction d x end cell row blank equals cell integral e to the power of 2 x end exponent d x end cell row blank equals cell e to the power of 2 x end exponent over 2 plus C end cell end table end style$

Kemudian, perhatikan perhitungan berikut,

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

Selanjutnya, perhatikan integral berikut.

Akan digunakan metode integral parsial untuk menyelesaikan integral tersebut.

Misalkan p = x dan $begin mathsize 14px style fraction numerator d v over denominator d x end fraction equals e to the power of 2 x end exponent end style$ , maka

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

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row q equals cell integral fraction numerator d q over denominator d x end fraction d x end cell row blank equals cell integral e to the power of 2 x end exponent d x end cell row blank equals cell e to the power of 2 x end exponent over 2 plus C end cell end table end style$

Sehingga diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral x squared e to the power of 2 x end exponent blank d x end cell equals cell fraction numerator x squared e to the power of 2 x end exponent over denominator 2 end fraction minus integral x e to the power of 2 x end exponent blank d x end cell row blank equals cell fraction numerator x squared e to the power of 2 x end exponent over denominator 2 end fraction minus open parentheses x times e to the power of 2 x end exponent over 2 minus integral e to the power of 2 x end exponent over 2 blank d x close parentheses end cell row blank equals cell fraction numerator x squared e to the power of 2 x end exponent over denominator 2 end fraction minus open parentheses fraction numerator x e to the power of 2 x end exponent over denominator 2 end fraction minus integral e to the power of 2 x end exponent over 2 blank d x close parentheses end cell row blank equals cell fraction numerator x squared e to the power of 2 x end exponent over denominator 2 end fraction minus open parentheses fraction numerator x e to the power of 2 x end exponent over denominator 2 end fraction minus e to the power of 2 x end exponent over 4 close parentheses plus C end cell row blank equals cell fraction numerator x squared e to the power of 2 x end exponent over denominator 2 end fraction minus fraction numerator x e to the power of 2 x end exponent over denominator 2 end fraction plus e to the power of 2 x end exponent over 4 plus C end cell row blank equals cell e to the power of 2 x end exponent over 4 open parentheses 2 x squared minus 2 x plus 1 close parentheses plus C end cell end table end style$

Jadi, jawaban yang tepat adalah B.