不定積分∫dx/(x^4 1)的計算步驟
本題通過湊分、換元、裂項、反正切函數導數、幂函數導數等方法和知識，介紹不定積分∫dx/(x^4 1)的主要計算步驟。
∫dx/(x^4 1)
=∫dx/(x^4 1)
=(1/2)∫[(x^2 1)-(x^2-1)]dx/(x^4 1)，此步驟為對分子進行等量變換，
=(1/2)∫(x^2 1)dx/(x^4 1)- (1/2)∫(x^2-1)dx/(x^4 1)，此步驟為裂項，
=(1/2)∫(x^2 1)dx/(x^4 1)- (1/2)∫(x^2-1)dx/(x^4 1)，兩項分子分母同時除以t^2得，
=(1/2)∫[1 (1/x^2)]dx/[x^2 (1/x^2)]- (1/2)∫[1-(1/x^2)]dx/[x^2 (1/x^2)]，
=(1/2)∫d(x-1/x)/[x^2 (1/x^2)]- (1/2)∫d(x 1/x)/[x^2 (1/x^2)]，
此步驟為分子湊分法，
=(1/2)∫d(x-1/x)/[(x-1/x)^2 2]-(1/2)∫d(x 1/x)/[(x 1/x)^2-2],此步驟為根據分子對分母進行配方計算，
=(1/2)∫d(x-1/x)/2[(x-1/x)^2/2 1]-(1/2)∫d(x 1/x)/{[(x 1/x)-√2][ (x 1/x) √2]},
此步驟前者對分母提取公因式2，後者使用平方差公式，即：
=(1/2)arctan[(x-1/x)/√2]- (1/4√2){∫d(x 1/x)/[(x 1/x)-√2]-∫d(x 1/x)/[(x 1/x) √2]},
=(1/2)arctan[(x-1/x)/√2]- (1/4√2)ln|[(x 1/x)-√2]/ [(x 1/x) √2]| C.
進行等量變形，則：
所求式
=(1/2)arctan[(x^2-1)/√2x]-(1/4√2)ln|[(x^2 1)-√2x]/ [(x^2 1) √2x]| C.
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