Pertanyaan

∫ x ( x + 1 ) 5 d x = ...

$\int x (x + 1)^{5} d x = ...$

D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Gunakan konsep integral parsial. Pilih dan . Perhatikan perhitungan berikut. Jadi, jawaban yang tepat adalah A.

Gunakan konsep integral parsial.

$integral u space straight d v equals u v minus integral v space straight d u integral k open parentheses a x plus b close parentheses to the power of n d x equals k over a times fraction numerator 1 over denominator n plus 1 end fraction open parentheses a x plus b close parentheses to the power of n plus 1 end exponent plus c$

Pilih u equals x dan straight d v equals open parentheses x plus 1 close parentheses to the power of 5 space straight d x .

$u equals x rightwards arrow fraction numerator straight d u over denominator straight d x end fraction equals 1 rightwards arrow straight d u equals straight d x$

$table attributes columnalign right center left columnspacing 2px end attributes row cell straight d v end cell equals cell open parentheses x plus 1 close parentheses to the power of 5 space straight d x end cell row v equals cell integral d v end cell row blank equals cell integral open parentheses x plus 1 close parentheses to the power of 5 space straight d x end cell row blank equals cell 1 over 1 times fraction numerator 1 over denominator 5 plus 1 end fraction open parentheses x plus 1 close parentheses to the power of 5 plus 1 end exponent end cell row v equals cell 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 end cell end table$

Perhatikan perhitungan berikut.

$table attributes columnalign right center left columnspacing 2px end attributes row cell integral u space d v end cell equals cell u v minus integral v d u end cell row cell integral x open parentheses x plus 1 close parentheses to the power of 5 space straight d x end cell equals cell x times 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 minus integral 1 over 6 open parentheses x plus 1 close parentheses to the power of 6 space d x end cell row blank equals cell x over 6 open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 6 integral open parentheses x plus 1 close parentheses to the power of 6 space d x end cell row blank equals cell x over 6 open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 6 times fraction numerator 1 over denominator 6 plus 1 end fraction open parentheses x plus 1 close parentheses to the power of 6 plus 1 end exponent plus c end cell row blank equals cell x over 6 open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 42 open parentheses x plus 1 close parentheses to the power of 7 plus c end cell row blank equals cell x over 6 open parentheses x plus 1 close parentheses to the power of 6 minus fraction numerator open parentheses x plus 1 close parentheses over denominator 42 end fraction open parentheses x plus 1 close parentheses to the power of 6 plus c end cell row blank equals cell fraction numerator 7 x minus open parentheses x plus 1 close parentheses over denominator 42 end fraction open parentheses x plus 1 close parentheses to the power of 6 plus c end cell row blank equals cell fraction numerator open parentheses 6 x minus 1 close parentheses over denominator 42 end fraction open parentheses x plus 1 close parentheses to the power of 6 plus c end cell row cell integral x open parentheses x plus 1 close parentheses to the power of 5 space straight d x end cell equals cell 1 over 42 open parentheses x plus 1 close parentheses to the power of 6 open parentheses 6 x minus 1 close parentheses plus c end cell end table$

Jadi, jawaban yang tepat adalah A.