求不定积分∫(x^1/2)/(1-x^1/3)dx
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令u=x^(1/6),x=u^6,dx=6u^5 du
∴∫√x/(1-x^1/3) dx
=6∫u^8/(1-u²) du
用长除法:u^8=(1-u²)(-u^6-u⁴-u²-1)+1
u^8/(1-u²)=-u^6-u⁴-u²-1+1/(1-u²)
∴6∫u^8/(1-u²) du
=6∫(-u^6-u⁴-u²-1) du+6∫du/(1-u²)
=-6/7*u^7-5/5*u^5-6/3*u³-6u+3∫[1/(u+1)-1/(u-1)] du
=(-6/7)u^7-(6/5)u^5-2u³-6u+3ln|u+1|-3ln|u-1|+C
=(-6/7)(x^1/6)^7-(6/5)(x^1/6)^5-2(x^1/6)³-6x^1/6+3ln|x^1/6+1|-3ln|x^1/6-1|+C
=(-6/7)x^7/6-(6/5)x^5/6-2√x-6x^1/6+3ln|(x^1/6+1)/(x^1/6-1)|+C
=(-2/35)(x^1/6)[21(x^2/3)+35(x^1/3)+15x+105)]+3ln|[(x^1/6)+1]/[(x^1/6)-1]|+C
∴∫√x/(1-x^1/3) dx
=6∫u^8/(1-u²) du
用长除法:u^8=(1-u²)(-u^6-u⁴-u²-1)+1
u^8/(1-u²)=-u^6-u⁴-u²-1+1/(1-u²)
∴6∫u^8/(1-u²) du
=6∫(-u^6-u⁴-u²-1) du+6∫du/(1-u²)
=-6/7*u^7-5/5*u^5-6/3*u³-6u+3∫[1/(u+1)-1/(u-1)] du
=(-6/7)u^7-(6/5)u^5-2u³-6u+3ln|u+1|-3ln|u-1|+C
=(-6/7)(x^1/6)^7-(6/5)(x^1/6)^5-2(x^1/6)³-6x^1/6+3ln|x^1/6+1|-3ln|x^1/6-1|+C
=(-6/7)x^7/6-(6/5)x^5/6-2√x-6x^1/6+3ln|(x^1/6+1)/(x^1/6-1)|+C
=(-2/35)(x^1/6)[21(x^2/3)+35(x^1/3)+15x+105)]+3ln|[(x^1/6)+1]/[(x^1/6)-1]|+C
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