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intsqrt(tan x)dx

∫tan⁡xdx \int \sqrt{\tan x} d x

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Full solution

Q. ∫tan⁡xdx \int \sqrt{\tan x} d x
  1. Set u=tan⁡(x)u = \tan(x): Let's start by setting u=tan⁡(x)u = \tan(x), which means du=sec⁡2(x) dxdu = \sec^2(x) \, dx.
  2. Use trigonometric identity: Since sec⁡2(x)=1+tan⁡2(x)\sec^2(x) = 1 + \tan^2(x) and u=tan⁡(x)u = \tan(x), we can write sec⁡2(x)\sec^2(x) as 1+u21 + u^2.
  3. Simplify the expression: This simplifies to ∫(u121+u2) du\int(\frac{u^{\frac{1}{2}}}{1 + u^2}) \, du.
  4. Make a substitution: Let's attempt a substitution: let v=1+u2v = 1 + u^2, then dv=2u dudv = 2u \, du.
  5. Finalize the integral: Simplify the integral: ∫(12)⋅(u−12v) dv\int (\frac{1}{2}) \cdot (\frac{u^{-\frac{1}{2}}}{v}) \, dv.

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