(x^2-5)/(x+1)(x-2)^2的不定积分
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(x^2-5)/(x+1)(x-2)^2的不定积分
![(x^2-5)/(x+1)(x-2)^2的不定积分](/uploads/image/z/17332183-55-3.jpg?t=%28x%5E2-5%29%2F%28x%2B1%29%28x-2%29%5E2%E7%9A%84%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86)
分解部分分式:
设(x^2-5)/(x+1)(x-2)^2=a/(x+1)+b(x-2)+c/(x-2)^2
去分母:x^2-5=a(x-2)^2+b(x+1)(x-2)+c(x+1)
令x=-1,得:-4=a*9,得a=-4/9
令x=2,得:-1=c*3,得c=-1/3
令x=0,得:-5=4a-2b+c,得:b=(4a+c+5)/2=(-16/9-1/3+5)/2=13/9
因此不定积分=aln|x+1|+bln|x-2|-c/(x-2)+C
=-4/9*ln|x+1|+13/9*ln|x-2|+1/[3(x-2)]+C
设(x^2-5)/(x+1)(x-2)^2=a/(x+1)+b(x-2)+c/(x-2)^2
去分母:x^2-5=a(x-2)^2+b(x+1)(x-2)+c(x+1)
令x=-1,得:-4=a*9,得a=-4/9
令x=2,得:-1=c*3,得c=-1/3
令x=0,得:-5=4a-2b+c,得:b=(4a+c+5)/2=(-16/9-1/3+5)/2=13/9
因此不定积分=aln|x+1|+bln|x-2|-c/(x-2)+C
=-4/9*ln|x+1|+13/9*ln|x-2|+1/[3(x-2)]+C