题目

不定积分 ：∫ √x/√x- 3^√x dx

提问时间：2021-03-15

答案

∫ √x/(√x- 3^√x) dx
换元,x=t^6
=∫ t^3 / (t^3-t^2) d(t^6)
=∫ t^3(6t^5) / (t^3-t^2) dt
=6∫ t^6/(t-1) dt
=6∫ (t^6-1+1)/(t-1) dt
=6∫ (t^5+t^4+t^3+t^2+t+1) dt + 6∫ 1/(t-1) d(t-1)
=6(t^6/6+t^5/5+t^4/4+t^3/3+t^2/2+t)+6ln|t-1|+C
=t^6+6t^5/5+3t^4/2+2t^3+3t^2+6t+6ln|t-1|+C
=x+6x^(5/6)/5+3x^(2/3)/2+2x^(1/2)+3x^(1/3)+6x^(1/6)+6ln|x^(1/6)-1|+C
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