Pertanyaan

Carilah nilai integral berikut! ∫ x ( 2 − x ) 2 d x

Carilah nilai integral berikut!

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

A. Fatta

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Jawaban

nilai integral dari adalah .

nilai integral dari $integral fraction numerator open parentheses 2 minus x close parentheses squared over denominator square root of x end fraction d x$ adalah 8 square root of x minus 8 over 3 square root of x cubed end root plus 2 over 5 square root of x to the power of 5 end root plus c

8 square root of x minus 8 over 3 square root of x cubed end root plus 2 over 5 square root of x to the power of 5 end root plus c

Pembahasan

Perhatikan perhitungan berikut! Bentuk integral di atas dapat disederhanakan menjadi seperti berikut. Berdasarkan sifat integral , sehingga Dengan demikian, nilai integral dari adalah .

Perhatikan perhitungan berikut!

Bentuk integral di atas dapat disederhanakan menjadi seperti berikut.

Berdasarkan sifat integral $integral a x to the power of n d x equals fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus c$ , sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator open parentheses 2 minus x close parentheses squared over denominator square root of x end fraction d x end cell equals cell integral 4 x to the power of negative bevelled 1 half end exponent minus 4 x to the power of bevelled 1 half end exponent plus x to the power of bevelled 3 over 2 end exponent d x end cell row blank equals cell fraction numerator 4 over denominator begin display style bevelled 1 half end style end fraction x to the power of bevelled 1 half end exponent minus fraction numerator 4 over denominator begin display style bevelled 3 over 2 end style end fraction x to the power of bevelled 3 over 2 end exponent plus fraction numerator 1 over denominator begin display style bevelled 5 over 2 end style end fraction x to the power of bevelled 5 over 2 end exponent plus c end cell row blank equals cell 8 square root of x minus 8 over 3 square root of x cubed end root plus 2 over 5 square root of x to the power of 5 end root plus c end cell end table$

Dengan demikian, nilai integral dari $integral fraction numerator open parentheses 2 minus x close parentheses squared over denominator square root of x end fraction d x$ adalah 8 square root of x minus 8 over 3 square root of x cubed end root plus 2 over 5 square root of x to the power of 5 end root plus c .