Kalkulus II tentang fraksi rasional(fungsi rasional)

Kalkulus II tentang fraksi rasional(fungsi rasional)

Jawaban Terkonfirmasi

Jawab:

Penjelasan dengan langkah-langkah:

$misal~frac{x^3-1}{x^2(x-2)^2}=frac{A}{x}+frac{B}{x^2}+frac{C}{x-2}+frac{D}{(x-2)^2}\\frac{x^3-1}{x^2(x-2)^2}=frac{Ax(x-2)^2+B(x-2)^2+Cx^2(x-2)+Dx^2}{x^2(x-2)^2}\\frac{x^3-1}{x^2(x-2)^2}=frac{Ax^3-4Ax^2+4Ax+Bx^2-4Bx+4B+Cx^3-2Cx^2+Dx^2}{x^2(x-2)^2}\\frac{x^3-1}{x^2(x-2)^2}=frac{(A+C)x^3+(-4A+B-2C+D)x^2+(4A-4B)x+4B}{x^2(x-2)^2}\\\maka~diperoleh~:\4B=-1~->B=-frac{1}{4}\\4A-4B=0~->A=-frac{1}{4}\\A+C=1~->C=frac{5}{4}\\-4A+B-2C+D=0~->D=frac{7}{4}$

$maka~frac{x^3-1}{x^2(x-2)^2}=-frac{1}{4x}-frac{1}{4x^2}+frac{5}{4(x-2)}+frac{7}{4(x-2)^2}\\int {frac{x^3-1}{x^2(x-2)^2}} , dx\\=int {-frac{1}{4x}-frac{1}{4x^2}+frac{5}{4(x-2)}+frac{7}{4(x-2)^2}} , dx\\=-frac{1}{4}ln|x|+frac{x}{4}+frac{5}{4}ln|x-2|+frac{7}{4(2-x)}+C$