求根号x/(1-三次根号x)的不定积分根号x/(1-三次根号x)的不定积分如何求?
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![求根号x/(1-三次根号x)的不定积分根号x/(1-三次根号x)的不定积分如何求?](/uploads/image/z/10270963-19-3.jpg?t=%E6%B1%82%E6%A0%B9%E5%8F%B7x%2F%281-%E4%B8%89%E6%AC%A1%E6%A0%B9%E5%8F%B7x%29%E7%9A%84%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E6%A0%B9%E5%8F%B7x%2F%281-%E4%B8%89%E6%AC%A1%E6%A0%B9%E5%8F%B7x%29%E7%9A%84%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E5%A6%82%E4%BD%95%E6%B1%82%3F)
求根号x/(1-三次根号x)的不定积分根号x/(1-三次根号x)的不定积分如何求?
求根号x/(1-三次根号x)的不定积分
根号x/(1-三次根号x)的不定积分如何求?
求根号x/(1-三次根号x)的不定积分根号x/(1-三次根号x)的不定积分如何求?
设 x^(1/6)=t,则 x=t^6,dx=6t^5dt;
∫{√x/[1- ³√x]}dx=∫[t³/(1-t²)]*6t^5 dt=6∫[t^8/(1-t²)]dt=3∫[t^8/(t-1)]dt-3∫[t^8/(t+1)]dt
=3∫[(t-1)+1]²dt/(t-1) -3∫[(t+1)-1]^8dt/(t+1)
=3∫(t-1)^7dt+3∫8(t-1)^6dt+3∫28(t-1)^5dt+3∫56(t-1)^4dt+3∫70(t-1)³dt+3∫56(t-1)²dt+3∫28(t-1)dt+3∫8dt+3∫dt/(t-1)
-3∫(t+1)^7dt+3∫8(t+1)^6dt-3∫28(t+1)^5dt+3∫56(t+1)^4dt-3∫70(t+1)³dt+3∫56(t+1)²dt-3∫28(t+1)dt+3*∫8dt-3∫dt/(t+1) +C
=(3/8)[(t-1)^8-(t+1)^8]+(24/7)[(t-1)^7+(t+1)^7]+14[(t-1)^6-(t+1)^6]+(168/5)[(t-1)^5+(t+1)^5]
+(105/2)[(t-1)^4-(t+1)^4]+56[(t-1)³+(t+1)³]+42[(t-1)²-(t+1)²]+48t+3ln[(t-1)/(t+1)] +C
再将 t=x^(1/6) 换入即得最后积分结果;