1+X4次方括号分之一的积分怎么算

Question

1+X4次方括号分之一的积分怎么算

Answer 1

∫ 1/(x⁴+1) dx
=(1/2)∫ [(1+x²)+(1-x²)]/(x⁴+1) dx
=(1/2)∫ (1+x²)/(x⁴+1) dx + (1/2)∫ (1-x²)/(x⁴+1) dx
分子分母同除以x²
=(1/2)∫ (1/x²+1)/(x²+1/x²) dx + (1/2)∫ (1/x²-1)/(x²+1/x²) dx
将分子放到微分符号后
=(1/2)∫ 1/(x²+1/x²) d(x-1/x) - (1/2)∫ 1/(x²+1/x²) d(x+1/x)
=(1/2)∫ 1/(x²+1/x²-2+2) d(x-1/x) - (1/2)∫ 1/(x²+1/x²+2-2) d(x+1/x)
=(1/2)∫ 1/[(x-1/x)²+2] d(x-1/x) - (1/2)∫ 1/[(x+1/x)²-2] d(x+1/x)
=(√2/4)arctan[(x-1/x)/√2] - (√2/8)ln|(x+1/x-√2)/(x+1/x+√2)| + C
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