∫x^5/（x^4-1）dx求不定积分

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∫x^5/（x^4-1）dx求不定积分
∫x^5/（x^4-1）dx求不定积分

∫x^5/（x^4-1）dx求不定积分

∫x^5/(x^4-1)dx
=∫ [x + x/(x^4-1) ]dx
let
x/(x^4-1) ≡ A/(x-1) +B/(x+1) + (Cx+D)/(x^2+1)
=>
x≡ A(x+1)(x^2+1) +B(x-1)(x^2+1) + (Cx+D)(x^2-1)
x=1
A(1+1)(1+1) =1
A = 1/4

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∫x^5/(x^4-1)dx
=∫ [x + x/(x^4-1) ]dx
let
x/(x^4-1) ≡ A/(x-1) +B/(x+1) + (Cx+D)/(x^2+1)
=>
x≡ A(x+1)(x^2+1) +B(x-1)(x^2+1) + (Cx+D)(x^2-1)
x=1
A(1+1)(1+1) =1
A = 1/4
x=-1
B(-1-1)(1+1) = -1
B =1/4
coef. of constant
A-B-D =0
D=0
coef. of x
A+B-C=1
C= -1/2
ie
∫x^5/(x^4-1)dx
=∫ [x + x/(x^4-1) ]dx
=∫ {x + (1/4)[1/(x-1)] +(1/4)[1/(x+1)] -(1/2)[x/(x^2+1)] }dx
= x^2/2 + (1/4)ln[(x-1)(x+1)] -(1/2)∫x dx/(x^2+1)
= x^2/2 + (1/4)ln[(x-1)(x+1)] -(1/4)ln(x^2+1) + C
= x^2/2 + (1/4)ln|(x^2-1)/(x^2+1)| + C

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