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Carilah  nilai integral berikut!

integral fraction numerator open parentheses 2 minus x close parentheses squared over denominator square root of x end fraction d x

A. Fatta

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

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Jawaban

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.

Pembahasan

Perhatikan perhitungan berikut!

Bentuk integral di atas dapat disederhanakan menjadi seperti berikut.

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 fraction numerator 4 minus 4 x plus x squared over denominator square root of x end fraction d x end cell row blank 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 end table

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.

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