Pertanyaan

∫ ( x − 2 ) 3 2 x + 5 d x

$\int \frac{2 x + 5}{( x - 2 ) ^{3}} d x$

R. Hajrianti

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x plus 5 over denominator open parentheses x minus 2 close parentheses cubed end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator open parentheses x minus 2 close parentheses end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 9 over denominator 2 open parentheses x minus 2 close parentheses squared end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table$ .

Pembahasan

Misal , maka diperoleh: Kemudian, dengan menggunakan metode substitusi, sebagai berikut. Dengan demikian, .

Misal u equals x minus 2 , maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell x minus 2 end cell row x equals cell u plus 2 end cell end table

$table attributes columnalign right center left columnspacing 0px end attributes row u equals cell x minus 2 end cell row cell fraction numerator straight d u over denominator straight d x end fraction end cell equals 1 row cell straight d u end cell equals cell straight d x end cell end table$

Kemudian, dengan menggunakan metode substitusi, $integral fraction numerator 2 x plus 5 over denominator open parentheses x minus 2 close parentheses cubed end fraction space straight d x$ sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator 2 x plus 5 over denominator open parentheses x minus 2 close parentheses cubed end fraction space straight d x end cell equals cell integral fraction numerator 2 open parentheses u plus 2 close parentheses plus 5 over denominator u cubed end fraction space straight d u end cell row blank equals cell integral fraction numerator 2 u plus 4 plus 5 over denominator u cubed end fraction space straight d u end cell row blank equals cell integral fraction numerator 2 u plus 9 over denominator u cubed end fraction space straight d u end cell row blank equals cell integral open parentheses 2 u plus 9 close parentheses u to the power of negative 3 end exponent space straight d u end cell row blank equals cell integral open parentheses 2 u times u to the power of negative 3 end exponent plus 9 u to the power of negative 3 end exponent close parentheses space straight d u end cell row blank equals cell integral open parentheses 2 u to the power of negative 2 end exponent plus 9 u to the power of negative 3 end exponent close parentheses space straight d u end cell row blank equals cell 2 times fraction numerator 1 over denominator negative 2 plus 1 end fraction times u to the power of negative 2 plus 1 end exponent plus 9 times fraction numerator 1 over denominator negative 3 plus 1 end fraction u to the power of negative 3 plus 1 end exponent plus C end cell row blank equals cell 2 times fraction numerator 1 over denominator open parentheses negative 1 close parentheses end fraction times u to the power of negative 1 end exponent plus 9 times fraction numerator 1 over denominator open parentheses negative 2 close parentheses end fraction times u to the power of negative 2 end exponent plus C end cell row blank equals cell negative 2 u to the power of negative 1 end exponent minus 9 over 2 u to the power of negative 2 end exponent plus C end cell row blank equals cell negative 2 over u minus fraction numerator 9 over denominator 2 u squared end fraction plus C end cell row blank equals cell negative fraction numerator 2 over denominator open parentheses x minus 2 close parentheses end fraction minus fraction numerator 9 over denominator 2 open parentheses x minus 2 close parentheses squared end fraction plus C end cell end table$

Dengan demikian, $table attributes columnalign right center left columnspacing 0px end attributes row blank blank integral end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x plus 5 over denominator open parentheses x minus 2 close parentheses cubed end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator open parentheses x minus 2 close parentheses end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 9 over denominator 2 open parentheses x minus 2 close parentheses squared end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank C end table$ .