Решите неопределенный интеграл
[tex] \int\limits \frac{dx}{\sqrt[3]{3x+1}-1} [/tex]
[tex] \int\limits \frac{dx}{\sqrt[3]{3x+1}-1} [/tex]
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∫[1/(∛(3x+1)-1)]dx=введем новую переменную 3x+1= t³ , тогда 3dx=3t²dt
или dx=t²dt , интеграл примет вид
∫[1/(t-1)]t²dt = ∫[t²/(t-1)]dt= ∫[(t²-1+1)/(t-1)]dt= ∫[(t²-1)/(t-1)]dt+ ∫[1/(t-1)]dt=
∫(t+1)dt+ ∫[1/(t-1)]dt=t²/2+t+lnIt-1I+C=∛(3x+1)²/2+∛(3x+1)+lnI∛(3x+1)-1I+C