1-x-1-3-1-dx-

Question Number 135633 by metamorfose last updated on 14/Mar/21

\int \frac{1}{x^{\frac{1}{3}} + 1} d x = \dots ?

Answered by Ñï= last updated on 14/Mar/21

\int \frac{d x}{x^{\frac{1}{3}} + 1} \overset{t = x^{\frac{1}{3}}}{=} \int \frac{3 t^{2} d t}{t + 1} = 3 \int \frac{(t + 1) (t - 1) + 1}{t + 1} d t = 3 \int {(t - 1) + \frac{1}{t + 1}} d t

= \frac{3}{2} t^{2} - 3 t + 3 l n (t + 1) + C = \frac{3}{2} x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 3 l n (1 + x^{\frac{1}{3}}) + C

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