Question

求助不定积分

算了很久都没算出来，哪位高手帮帮我啊！

Answer 1

代u=x^1/6,x=u^6,dx=6u^5 du
∫√x/(√x-x^1/3) dx
=6∫u^8/(u³-u²) du
=6∫u^6/(u-1) du
用长除法解u^6/(u-1)得
u^6=(u-1)(u^5+u⁴+u³+u²+u+1)+1
u^6/(u-1)=u^5+u⁴+u³+u²+u+1+1/(u-1)
∴6∫u^6/(u-1) du
=6∫(u^5+u⁴+u³+u²+u+1) du+6∫du/(u-1)
=6(1/6*u^6+1/5*u^5+1/4*u⁴+1/3*u³+1/2*u²+u)+6ln|u-1|+C
=u^6+(6/5)u^5+(3/2)u⁴+2u³+3u²+6u+6ln|u-1|+C
=x+(6/5)[x^(5/6)]+(3/2)[x^(2/3)]+2√x+3x^(1/3)+6x^(1/6)+6ln|x^(1/6)-1|+C

Answer 2

*(35-.xlo)8/3.5-f>=(.00.2*uy.3/n3631)&hsy=op3.65

Answer 3

可令x=t^6
原式=∫6t^6/(t-1)dt
u=t-1
原式=∫6(u+1)^6/udu
把(u+1)^6展开，分成多项的和 再求积分就简单了
记得把u代回t，t再代回x