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∫ ( 3 x 2 + x 3 2 ​ + x 2 4 ​ ) d x = ...

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R. Hajrianti

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

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 integral open parentheses 3 x squared plus 2 over x cubed plus 4 over x squared close parentheses space straight d x equals x cubed minus 1 over x squared minus 4 over x plus C.

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Pembahasan

Jika , maka . Lalu, salah satu sifat integral tak tentu adalah . Dengan menggunakan rumus pengintegralan dan sifat integral tersebut, nilai sebagai berikut. Dengan demikian, .

Jika f open parentheses x close parentheses equals x to the power of n, maka integral f open parentheses x close parentheses space straight d x equals fraction numerator 1 over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus C. Lalu, salah satu sifat integral tak tentu adalah integral open parentheses f open parentheses x close parentheses plus g open parentheses x close parentheses close parentheses space straight d x equals integral f open parentheses x close parentheses space straight d x plus integral g open parentheses x close parentheses space straight d x. Dengan menggunakan rumus pengintegralan dan sifat integral tersebut, nilai integral open parentheses 3 x squared plus 2 over x cubed plus 4 over x squared close parentheses space straight d x sebagai berikut.
 

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


Dengan demikian, integral open parentheses 3 x squared plus 2 over x cubed plus 4 over x squared close parentheses space straight d x equals x cubed minus 1 over x squared minus 4 over x plus C.

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