Question

求不定积分，谢谢！

Answer 1

这题有点复杂
设u=x^(1/6),x=u^6,dx=6u^5 du
∫√x/(1+³√x) dx
=6∫u^8du/(1+u²)
=6∫(u^8+u^6-u^6)du/(u²+1)
=6∫u^6du-6∫u^6du/(u²+1)
=6∫u^6du-6∫(u^6+u^4-u^4)du/(u²+1)
=6∫u^6du-6∫u^4du+6∫u^4du/(u²+1)
=6∫u^6du-6∫u^4du+6∫(u^4+u²-u²)du/(u²+1)
=6∫u^6du-6∫u^4du+6∫u²du-6∫u²du/(u²+1)
=6∫u^6du-6∫u^4du+6∫u²du-6∫(u²+1-1)du/(u²+1)
=6∫u^6du-6∫u^4du+6∫u²du-6∫du+6∫du/(u²+1)
=6[u^7/7-u^5/5+u³/3-u+arctanu]+C
再把u=(x)^(1/6)代入即可。