Question

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

Answer 1

`A(x)=3ln|root(3)x+1|+"c"`

Explanation:

`int 1/(x^(2/3)(1+x^(1/3))` `dx`

`=3int x^(-2/3)/(3(1+x^(1/3))` `dx`

We now have the integrand in the form `(f'(x))/f(x)`. Using the reverse chain rule, we know that the integrals of these forms are `ln|f(x)|+"c"`.

`therefore int 1/(x^(2/3)(1+x^(1/3))` `dx=3ln|root(3)x+1|+"c"`