∫1/(x^4-1)dx1/(x^4-1)为被积函数,急用最好在2天内解决,越快

Question

∫1/(x^4-1)dx
1/(x^4-1)为被积函数,急用最好在2天内解决,越快

Answer 1

1/(x^4-1)
=1/(x²+1)(x²-1)
=1/2*[1/x²-1)-1/x²+1)]
同理1/(x²-1)=1/2*[1/(x-1)-1/(x+1)]
所以1/(x^4-1)
=(1/4)*1/(x-1)-(1/4)*1/(x+1)-(1/2)*1/(x²+1)
所以原式=1/4∫dx/(x-1)-1/4∫dx/(x+1)-1/2∫dx/(x²+1)
=1/4∫d(x-1)/(x-1)-1/4∫d(x+1)/(x+1)-1/2∫dx/(x²+1)
=1/4*ln|x-1|-1/4*ln|x+1|-1/2*arctanx+C
=1/4*ln|(x-1)/(x+1)|-1/2*arctanx+C