What is the integral of \sqrt{\tan(x)} and what is the reasoning behind it?

 What is the integral of \sqrt{\tan(x)} and what is the reasoning behind it?👇🏾👇🏾

The integral of $\sqrt{\tan (𝑥)}$ with respect to $𝑥$ is a non-trivial integral that does not have a simple closed-form solution. To express the integral and understand the reasoning behind it, we can use a substitution method to simplify the integral. Here is the step-by-step approach:

1. Substitution: Let $𝑢=\tan (𝑥)$. Then, the differential $𝑑𝑢={\sec }^2(𝑥)𝑑𝑥$.

2. Relating $𝑑𝑥$ to $𝑑𝑢$: Since $𝑑𝑢={\sec }^2(𝑥)𝑑𝑥$ and knowing that ${\sec }^2(𝑥)=1+{\tan }^2(𝑥)$, we can express $𝑑𝑥$ as follows:

   $$𝑑𝑥=\frac{𝑑𝑢}{{\sec }^2(𝑥)}=\frac{𝑑𝑢}{1+{𝑢}^2}$$

   because $𝑢=\tan (𝑥)$.

3. Substitute into the integral: We now substitute $𝑢=\tan (𝑥)$ and $𝑑𝑥=\frac{𝑑𝑢}{1+{𝑢}^2}$ into the original integral:

   $$\int \sqrt{\tan (𝑥)}𝑑𝑥=\int \sqrt{𝑢}\cdot \frac{𝑑𝑢}{1+{𝑢}^2}$$

4. Integrate by parts or other advanced techniques: This integral, $\int \frac{\sqrt{𝑢}}{1+{𝑢}^2}𝑑𝑢$, is still complex and does not simplify easily using elementary functions. It can be expressed in terms of special functions or through numerical integration methods. Specifically, it can be expressed in terms of the incomplete Beta function or the hypergeometric function after certain transformations.

Thus, the integral of $\sqrt{\tan (𝑥)}$ is not straightforward and does not have a simple antiderivative in terms of elementary functions. For practical purposes, the integral might be evaluated numerically or approximated using series expansions or special functions in more advanced settings.

To sum up, the integral of $\sqrt{\tan (𝑥)}$ is:

$$\int \sqrt{\tan (𝑥)}𝑑𝑥$$

is a complex integral that typically requires special functions or numerical methods for its evaluation. It does not have a simple closed-form solution in terms of elementary functions.