计算不定积分∫dx/∛(x )+√(x ))

计算不定积分∫dx/∛(x )+√(x ))
计算不定积分∫dx/∛(x )+√(x ))

计算不定积分∫dx/∛(x )+√(x ))
令a=x^(1/6)
则x=a^6
dx=6a^5da
则分母=a²+a³
原式=∫6a^5da/(a²+a³)
=6∫a³da/(a+1）
=6∫(a³+1-1)da/(a+1）
=6∫[(a+1)(a²-a+1)-1]da/(a+1）
=6∫[(a²-a+1-1/(a+1]da
==6[a³/3-a²/2+a-ln(a+1)]+C
=2√x-3³√x+x^(1/6)-ln[x^(1/6)+1]+C

x=t^6变换
∫dx/∛(x )+√(x ))=∫6*t^5dt/(t^2+t^3)
=∫6*t^3dt/(1+t)=x^(1/6) - 3x^(1/3) + 2x^(1/2) - 6Ln[1 + x^(1/6)]+C