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integral fraction numerator s space d s over denominator square root of s plus 3 end root end fraction

Jawaban:

Pertanyaan dari soal tersebut adalah menentukan hasil dari integral integral fraction numerator s space d s over denominator square root of s plus 3 end root end fraction. Kita gunakan konsep integral substitusi.

Rumus umum integral (anti turunan) sebagai berikut.

integral x to the power of n d x equals fraction numerator x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c

Kita buat permisalan dari integral tersebut. Misal,

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell s plus 3 left right arrow x minus 3 equals s end cell row cell d x end cell equals cell d s end cell end table

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell integral fraction numerator s space d s over denominator square root of s plus 3 end root end fraction end cell equals cell integral fraction numerator x minus 3 space d x over denominator square root of x end fraction end cell row blank equals cell integral fraction numerator x over denominator square root of x end fraction d x minus integral fraction numerator 3 over denominator square root of x end fraction d x end cell row blank equals cell integral x to the power of 1 half end exponent d x minus integral 3 x to the power of negative 1 half end exponent d x end cell row blank equals cell fraction numerator x to the power of begin display style 1 half end style plus 1 end exponent over denominator begin display style 1 half end style plus 1 end fraction minus fraction numerator 3 x to the power of negative begin display style 1 half end style plus 1 end exponent over denominator negative begin display style 1 half end style plus 1 end fraction end cell row blank equals cell fraction numerator x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction minus fraction numerator 3 x to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction end cell row blank equals cell 2 over 3 x to the power of 3 over 2 end exponent minus 6 x to the power of 1 half end exponent plus c end cell end table

Kemudian kita kembalikan ke permisalannya menjadi 2 over 3 open parentheses s plus 3 close parentheses to the power of 3 over 2 end exponent minus 6 open parentheses s plus 3 close parentheses to the power of 1 half end exponent plus c.

Jadi, integral fraction numerator s space d s over denominator square root of s plus 3 end root end fraction equals 2 over 3 open parentheses s plus 3 close parentheses to the power of 3 over 2 end exponent minus 6 open parentheses s plus 3 close parentheses to the power of 1 half end exponent plus c.

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