求一些积分？高手进

求一些积分？高手进

f(x)=x^n+1=0

x^n=-1=e^i(2k+1)π

x有n个复数解为x1=e^iπ/n x2=e^i3π/n .........xn=e^i(2n-1)π/n

x^n+1=(x-e^iπ/n ).......(x-e^i(2n-1)π/n)

1/(x^n+1)=[1/f'(x1)]/(x-xn )+.......+[1/f'(xn)]/(x-xn)

∫1/(1+x^n)dx=∑1/[1/f'(xk)]ln(x-xk)+c k从1加到n

验证n=2时∫1/(1+x^2)dx=arctanx+c

x^2+1=0 x1=e^iπ/2 x2=e^i3π/2

∫1/(1+x^n)dx=1/(e^iπ/2-e^i3π/2)ln(x-e^iπ/2)-1/(e^iπ/2-e^i3π/2)ln(x-e^i3π/2)

=1/(e^iπ/2-e^i3π/2)ln(x-e^iπ/2)/(x-e^i3π/2)=-i/2ln(x-i)/ln(x+i)=arctanx

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