x^2/(x-1)^10 dx求不定积分详细过程 急

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x^2/(x-1)^10 dx求不定积分详细过程 急

x^2/(x-1)^10 dx求不定积分详细过程 急
x^2/(x-1)^10 dx求不定积分详细过程 急

x^2/(x-1)^10 dx求不定积分详细过程 急
x^2/(x-1)^10
说明：x^2=(x-1+1)^2=(x-1)^2+2(x-1)+1
x^2/(x-1)^10
=((x-1)^2+2(x-1)+1)/(x-1)^10
=1/(x-1)^8+(2(x-1)+1)/(x-1)^10
=(x-1)^(-8) +2 (x-1)^(-9) +(x-1)^(-10)
原式=∫[(x-1)^(-8) +2 (x-1)^(-9) +(x-1)^(-10)]d(x-1)
=-[1/[7(x-1)^7]-[2/[8(x-1)^(-8)]-[1/[9(x-1)^(-9)] + C
=-1/[7(x-1)^7]-1/[4(x-1)^8]-1/[9(x-1)^9]+C

令x-1=t
∫x^2/(x-1)^10 dx

=∫（t+1）²/t^10dt
=∫（1/t^8+2/t^9+1/t^10）dt
=-1/7t^7-1/4t^8-1/9t^9+C
=-1/[7(x-1)^7]-1/[4(x-1)^8]-1/[9(x-1)^9]+C


∫x^2/(x-1)^10 dx
=∫[1/(x-1)^8+2/(x-1)^9+1/(x-1)^10]dx
=∫1/(x-1)^8 dx+2∫1/(x-1)^9 dx+∫1/(x-1)^10 dx
=∫1/(x-1)^8 d(x-1)+2∫1/(x-1)^9 d(x-1)+∫1/(x-1)^10 d(x-1)
=(-1/7)·1/(x-1)^7+(-2/8)·1/(x-1)^8+(-1/9)·1/(x-1)^9+C
=-1/[7(x-1)^7]-1/[4(x-1)^8]-1/[9(x-1)^9]+C

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