If f(x) has an inverse on [a, b], then integrating by parts (take u = f(x) and dv = dx), we can show

Let
. Compute the inverse:
![f\left(f^(-1)(x)\right) = \sqrt{1 + f^(-1)(x)^3} = x \implies f^(-1)(x) = \sqrt[3]{x^2-1}](https://img.qammunity.org/2023/formulas/mathematics/college/5vgiw52ss1rgifdcfyr8b8e1ycz890eea8.png)
and we immediately notice that
.
So, we can write the given integral as

Splitting up terms and replacing
in the first integral, we get
