Pertanyaan

Tentukan integral berikut dengan integral parsial b. ∫ x 3 ( x + 1 ) 3 d x

Tentukan integral berikut dengan integral parsial

b. $\int x^{3} (x + 1)^{3} d x$

S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Misalkan dan maka: Dengan menggunakan konsepintegral parsial maka: Kemudian tentukan integral dari , ikuti langkah langkah di atas sehingga diperoleh: maka: Kemudian tentukan integral dari : maka: kemudian substitusikan ke maka: Dengan demikian hasil dari adalah

Misalkan u equals x cubed dan d v equals open parentheses x plus 3 close parentheses cubed maka:

$table row cell u equals x cubed end cell rightwards arrow cell fraction numerator d u over denominator d x end fraction equals 3 x squared end cell row blank blank cell d u equals 3 x squared space d x end cell row cell d v equals open parentheses x plus 1 close parentheses cubed end cell rightwards arrow cell v equals integral open parentheses x plus 1 close parentheses cubed space d x end cell row blank blank cell v equals 1 fourth open parentheses x plus 1 close parentheses to the power of 4 space end cell end table$

Dengan menggunakan konsep integral parsial maka:

Kemudian tentukan integral dari , ikuti langkah langkah di atas sehingga diperoleh:

maka:

Kemudian tentukan integral dari integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x :

maka:

kemudian substitusikan ke maka:

$begin mathsize 12px style integral x cubed open parentheses x plus 1 close parentheses cubed space d x equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus 1 fourth integral open parentheses x plus 1 close parentheses to the power of 4 times 3 x squared space d x equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus 1 fourth open square brackets fraction numerator 3 x squared over denominator 5 end fraction open parentheses x plus 1 close parentheses to the power of 5 minus 1 fifth integral open parentheses x plus 1 close parentheses to the power of 5 times 6 x space d x close square brackets equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus fraction numerator 3 x squared over denominator 20 end fraction open parentheses x plus 1 close parentheses to the power of 5 plus 1 over 20 open square brackets fraction numerator 6 x over denominator 6 end fraction open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 7 open parentheses x plus 1 close parentheses to the power of 7 close square brackets equals x cubed over 4 open parentheses x plus 1 close parentheses to the power of 4 minus fraction numerator 3 x squared over denominator 20 end fraction open parentheses x plus 1 close parentheses to the power of 5 plus x over 20 open parentheses x plus 1 close parentheses to the power of 6 minus 1 over 140 open parentheses x plus 1 close parentheses to the power of 7 end style$

Dengan demikian hasil dari integral x cubed open parentheses x plus 1 close parentheses cubed space d x adalah

$size 12px x to the power of size 12px 3 over size 12px 4 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 4 size 12px minus fraction numerator size 12px 3 size 12px x to the power of size 12px 2 over denominator size 12px 20 end fraction begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 5 size 12px plus size 12px x over size 12px 20 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 6 size 12px minus size 12px 1 over size 12px 140 begin mathsize 12px style left parenthesis x plus 1 right parenthesis end style to the power of size 12px 7$