Pertanyaan

Nilai dari ∫ 3 x ( 2 − x ) 4 d x adalah ....

Nilai dari $\int 3 x (2 - x)^{4} d x$ adalah ....

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Kita cari hasil integral di atas dengan menggunakan integral parsial. Misalkan dan , maka dan sehingga Jadi, jawaban yang tepat adalah E.

Kita cari hasil integral di atas dengan menggunakan integral parsial.
Misalkan dan $begin mathsize 14px style fraction numerator straight d v over denominator straight d x end fraction equals left parenthesis 2 minus x right parenthesis to the power of 4 end style$ , maka

$begin mathsize 14px style fraction numerator straight d u over denominator straight d x end fraction equals 3 end style$

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row v equals cell integral fraction numerator straight d v over denominator straight d x end fraction straight d x end cell row blank equals cell integral left parenthesis 2 minus x right parenthesis to the power of 4 space straight d x semicolon space gunakan space metode space substitusi end cell row blank equals cell negative 1 fifth left parenthesis 2 minus x right parenthesis to the power of 5 plus C end cell end table end style$

sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral u straight d v end cell equals cell u v minus integral v straight d u end cell row cell integral 3 x left parenthesis 2 minus x right parenthesis to the power of 4 straight d x end cell equals cell 3 x open parentheses negative 1 fifth left parenthesis 2 minus x right parenthesis to the power of 5 close parentheses minus integral negative 1 fifth left parenthesis 2 minus x right parenthesis to the power of 5 times 3 straight d x end cell row blank equals cell negative 3 over 5 x left parenthesis 2 minus x right parenthesis to the power of 5 plus 3 over 5 integral left parenthesis 2 minus x right parenthesis to the power of 5 straight d x end cell row blank equals cell negative 3 over 5 x left parenthesis 2 minus x right parenthesis to the power of 5 plus 3 over 5 open parentheses negative 1 over 6 left parenthesis 2 minus x right parenthesis to the power of 6 close parentheses plus C end cell row blank equals cell negative 3 over 5 left parenthesis 2 minus x right parenthesis to the power of 5 open parentheses x plus fraction numerator 2 minus x over denominator 6 end fraction close parentheses plus C space end cell row blank equals cell negative 3 over 5 left parenthesis 2 minus x right parenthesis to the power of 5 open parentheses fraction numerator 6 x plus 2 minus x over denominator 6 end fraction close parentheses plus C end cell row blank equals cell negative 3 over 5 left parenthesis 2 minus x right parenthesis to the power of 5 open parentheses fraction numerator 2 plus 5 x over denominator 6 end fraction close parentheses plus C end cell row blank equals cell negative 3 over 30 left parenthesis 2 minus x right parenthesis to the power of 5 left parenthesis 2 plus 5 x right parenthesis plus C end cell row blank equals cell negative 1 over 10 left parenthesis 2 minus x right parenthesis to the power of 5 left parenthesis 2 plus 5 x right parenthesis plus C end cell end table end style$

Jadi, jawaban yang tepat adalah E.