Question

$\int \frac{x+1}{x^{\frac{1}{3}}+1}dx$ is equal to
(where $C$ is constant of integration)

Answer 1

$\Rightarrow \int \frac{(x^{\frac{1}{3}}+1)(x^{\frac{2}{3}}+1-x^{\frac{1}{3}})}{x^{\frac{1}{3}}+1}dx\{\because x+1={\left(x^{\frac{1}{3}}\right)}^3+{\left(1^{\frac{1}{3}}\right)}^3=(x^{\frac{1}{3}}+1\left)\right(x^{\frac{2}{3}}+1-x^{\frac{1}{3}}\left)\right\}$
$\Rightarrow \int x^{\frac{2}{3}}dx+\int dx-\int x^{\frac{1}{3}}dx\Rightarrow \frac{x^{\frac{5}{3}}}{\frac{5}{3}}+x-\frac{x^{\frac{4}{3}}}{\frac{4}{3}}+C\Rightarrow \frac{3}{5}{x}^{\frac{5}{3}}+x-\frac{3}{4}{x}^{\frac{4}{3}}+C$