Integration of √tanx

Introduction
The integration of √tanx can be a tricky process, but it can be simplified by using substitution and trigonometric identities. In this guide, we will provide step-by-step instructions on how to integrate √tanx.

Steps

1. Use the substitution u = tan x, then du/dx = sec²x dx.
2. Rewrite the integral in terms of u: ∫√tanx dx = ∫√u * (1/ cos²x) dx.
3. Substitute cos²x = (1 + tan²x) and simplify: ∫√u * (1/ 1 + u) du.
4. Use the substitution v = √u, then dv/dx = (1/2√u) du.
5. Rewrite the integral in terms of v: 2∫ v² / (1 + v²) dv.
6. Use the substitution w = 1 + v², then dw/dv = 2v. Rewrite the integral in terms of w: ∫(w - 1)/w dw.
7. Split the integral into two parts: ∫dw/w - ∫dw/(w + 1).
8. Simplify the integrals: ln|w| - ln|w + 1|.
9. Substitute back in terms of u and v: ln|√u| - ln|√(1 + u)|.
10. Substitute back in terms of x: ln|√tanx| - ln|√(1 + tanx)|.

Conclusion
The integration of √tanx can be a challenging process, but by using substitution and trigonometric identities, it can be simplified. Following the steps outlined in this guide will allow you to integrate √tanx with ease.