求x^5-x/x^8+1dx的积分.

Question

求x^5-x/x^8+1dx的积分.

Answer 1

x^5-x=x(x²+1)(x²-1)
x^8+1=(x²+1)(x^4-x²+1)
约去(x²+1)原积分化为：
∫ (x³-x)/(x^4-x²+1)dx
=(1/4)∫ (4x³-2x-2x)/(x^4-x²+1)dx
=(1/4)∫ (4x³-2x)/(x^4-x²+1)dx-(1/4)∫ 2x/(x^4-x²+1)dx
=(1/4)∫ 1/(x^4-x²+1)d(x^4-x²) - (1/4)∫ 1/(x^4-x²+1)d(x²)
注意到：x^4-x²+1=(x²-1/2)²+3/4
=(1/4)ln(x^4-x²+1) - (1/4)∫ 1/[(x²-1/2)²+3/4]d(x²)
=(1/4)ln(x^4-x²+1) - (1/4)(2/√3)arctan[2(x²-1/2)/√3]+C
=(1/4)ln(x^4-x²+1) - (1/(2√3))arctan[2(x²-1/2)/√3]+C
用数学软件求导验算,结果正确.