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Integrate tan⁡x dx\sqrt{\tan x}\, dx

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Full solution

Q. Integrate tan⁡x dx\sqrt{\tan x}\, dx
  1. Identify integral: Identify the integral we need to solve: ∫tan⁡(x) dx\int\sqrt{\tan(x)} \, dx.
  2. Let uu substitution: Let u=tan⁡(x)u = \tan(x), which means dudx=sec⁡2(x)\frac{du}{dx} = \sec^2(x) or dx=dusec⁡2(x)dx = \frac{du}{\sec^2(x)}.
  3. Substitute uu and dxdx: Substitute uu and dxdx in the integral: ∫u⋅dusec⁡2(x)\int\sqrt{u} \cdot \frac{du}{\sec^2(x)}.
  4. Use trigonometric identity: Since sec⁡2(x)=1+tan⁡2(x)\sec^2(x) = 1 + \tan^2(x), we can write sec⁡2(x)\sec^2(x) as 1+u21 + u^2.
  5. Modify integral: Now the integral looks like ∫u/(1+u2) du\int \sqrt{u}/(1 + u^2) \, du.
  6. Perform integration: Perform the integration using a suitable method, like substitution or partial fractions.

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