∫1/（x^1/2+x^1/3）谢谢了不是很会做答案是2x^1/2-3x^1/3+6x^1/6-6ln(1+6x^1/6)+c

Question

∫1/（x^1/2+x^1/3）谢谢了不是很会做 答案是2x^1/2-3x^1/3+6x^1/6-6ln(1+6x^1/6)+c

Answer 1

此为典型题,思路是去掉所有根号,因此设x=t^6,则dx = 6t^5dt,原积分=∫ 6t^5dt/(t^2+t^3)
=∫6t^3dt/(1+t) =∫(6t^3-6+6)/(1+t)dt=6∫1-t+t^2)dt - 6∫1/(1+t)dt=6t-3t^2+2t^3-6ln|1+t| + C
将t=x^(1/6)带回上式,即得你给出的答案.