√x++³√x分之一求不定积分
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令 t = x^(1/6), 则 x = t^6, dx = 6t^5dt, 得
I = ∫dx/[√x + x^(1//3)] = ∫6t^5dt/(t^3 + t^2)
= 6∫t^3dt/(t+1) = 6∫(t^3+t^2-t^2-t+t+1-1)dt/(t+1)
= 6∫[t^2-t+1 -1/(t+1)]dt
= 6[t^3/3 - t^2/2 + t - ln(t+1)] + C
= 2√x - 3x^(1/3) + 6x^(1/6) - 6ln[x^(1/6)+1] + C
I = ∫dx/[√x + x^(1//3)] = ∫6t^5dt/(t^3 + t^2)
= 6∫t^3dt/(t+1) = 6∫(t^3+t^2-t^2-t+t+1-1)dt/(t+1)
= 6∫[t^2-t+1 -1/(t+1)]dt
= 6[t^3/3 - t^2/2 + t - ln(t+1)] + C
= 2√x - 3x^(1/3) + 6x^(1/6) - 6ln[x^(1/6)+1] + C
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