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Question Number 103846 by Dwaipayan Shikari last updated on 17/Jul/20

$$\int \frac{dx}{x^{\frac{1}{3}}+2}$$

Commented by Dwaipayan Shikari last updated on 17/Jul/20

$$\int \frac{3u^{2}du}{u+2}=\int \frac{3u^2-12}{u+2}+\frac{12}{u+2}=\int 3u-6+12log(x^{\frac{1}{3}}+2)=\frac{3}{2}{x}^{\frac{2}{3}}-6x^{\frac{1}{3}}+12log(x^{\frac{1}{3}}+2)+C$$

Commented by PRITHWISH SEN 2 last updated on 17/Jul/20

$$\mathrm{x}={\mathrm{t}}^3\Rightarrow \mathrm{dx}={3\mathrm{t}}^2\mathrm{dt}$$$$\int \frac{{3\mathrm{t}}^2\mathrm{dt}}{\mathrm{t}+2}=3\int \frac{{(\mathrm{t}+2)}^2-4(\mathrm{t}+2)+4}{(\mathrm{t}+2)}\mathrm{dt}$$$$=3\{\int (\mathrm{t}+2)\mathrm{dt}-4\int \mathrm{dt}+4\int \frac{\mathrm{dt}}{\mathrm{t}+2}$$

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