Pertanyaan

Hasil dari ∫ 3 x ( 6 − x ) 3 d x adalah ....

Hasil dari $\int 3 x (6 - x)^{3} d x$ adalah ....

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

hasil dari integral di atas adalah .

hasil dari integral di atas adalah negative 3 over 10 left parenthesis 2 x plus 3 right parenthesis left parenthesis 6 minus x right parenthesis to the power of 4 plus C

negative 3 over 10 left parenthesis 2 x plus 3 right parenthesis left parenthesis 6 minus x right parenthesis to the power of 4 plus C

Pembahasan

Diberikan bentuk integral . Karena terdiri dari perkalian dua fungsi maka dapat dimisalkan dan, Maka dengan metode integral parsial, penyelesaiannya adalah : Jadi, hasil dari integral di atas adalah .

Diberikan bentuk integral integral 3 x left parenthesis 6 minus x right parenthesis cubed space d x . Karena terdiri dari perkalian dua fungsi maka dapat dimisalkan

$table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 3 x end cell row cell fraction numerator d u over denominator d x end fraction end cell equals 3 end table$

dan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d v over denominator d x end fraction end cell equals cell left parenthesis 6 minus x right parenthesis cubed end cell row cell integral fraction numerator d v over denominator d x end fraction d x end cell equals cell integral left parenthesis 6 minus x right parenthesis cubed space d x end cell row v equals cell negative 1 fourth left parenthesis 6 minus x right parenthesis to the power of 4 end cell end table$

Maka dengan metode integral parsial, penyelesaiannya adalah :

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

Jadi, hasil dari integral di atas adalah negative 3 over 10 left parenthesis 2 x plus 3 right parenthesis left parenthesis 6 minus x right parenthesis to the power of 4 plus C .