We need to evaluate the integral:

\[
\int \frac{dx}{1 + \sqrt{\tan(x)}}
\]

To solve this, we can try a substitution to simplify the expression. Let:

\[
u = \sqrt{\tan(x)} \implies u^2 = \tan(x)
\]

Next, differentiate both sides of \( u^2 = \tan(x) \) with respect to \( x \):

\[
2u \frac{du}{dx} = \sec^2(x)
\]

From \( \tan(x) = u^2 \), we know that \( \sec^2(x) = 1 + u^2 \), so:

\[
2u \, du = (1 + u^2) \, dx
\]

Thus, the differential \( dx \) becomes:

\[
dx = \frac{2u \, du}{1 + u^2}
\]

Substitute this into the integral:

\[
\int \frac{\frac{2u \, du}{1 + u^2}}{1 + u} = 2 \int \frac{u \, du}{(1 + u^2)(1 + u)}
\]

This integral is still quite complex, but it can be tackled using partial fraction decomposition or other advanced techniques. Would you like to proceed with these more detailed steps?

Explore the step-by-step solution for integrating ∫dx/(1 + sqrt(tan(x))). This problem involves advanced calculus techniques, including substitution and partial fraction decomposition. Suitable for undergraduate students studying calculus.

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