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introot(5)(tan x)dx

∫tan⁡x5dx \int \sqrt[5]{\tan x} d x

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Full solution

Q. ∫tan⁡x5dx \int \sqrt[5]{\tan x} d x
  1. Identify integral: Identify the integral to be solved.\newlineWe need to solve ∫(tan⁡(x)15) dx\int(\tan(x)^{\frac{1}{5}}) \, dx.
  2. Apply substitution: Apply substitution if possible.\newlineLet u=tan⁡(x)u = \tan(x), then du=sec⁡2(x)dxdu = \sec^2(x) dx.
  3. Rewrite in terms of u: Rewrite the integral in terms of u.\newline∫(u15)⋅(1cos⁡2(x))dx\int(u^{\frac{1}{5}}) \cdot (\frac{1}{\cos^2(x)}) dx, but we need to express 1cos⁡2(x)\frac{1}{\cos^2(x)} in terms of u.

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