# How do you find the indefinite integral of int 1/(x^(2/3)(1+x^(1/3))?

May 19, 2017

$A \left(x\right) = 3 \ln | \sqrt[3]{x} + 1 | + \text{c}$

#### Explanation:

int 1/(x^(2/3)(1+x^(1/3)) $\mathrm{dx}$

=3int x^(-2/3)/(3(1+x^(1/3)) $\mathrm{dx}$

We now have the integrand in the form $\frac{f ' \left(x\right)}{f} \left(x\right)$. Using the reverse chain rule, we know that the integrals of these forms are $\ln | f \left(x\right) | + \text{c}$.

therefore int 1/(x^(2/3)(1+x^(1/3)) $\mathrm{dx} = 3 \ln | \sqrt[3]{x} + 1 | + \text{c}$