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int sec t(sec t+tan t)dt

∫sec⁡t(sec⁡t+tan⁡t)dt \int \sec t(\sec t+\tan t) d t =

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Full solution

Q. ∫sec⁡t(sec⁡t+tan⁡t)dt \int \sec t(\sec t+\tan t) d t =
  1. Distribute sec⁡(t)\sec(t): Rewrite the integral by distributing sec⁡(t)\sec(t) inside the parentheses.\newlineCalculation: \int \sec(t)(\sec(t) + \tan(t)) \, dt = \int (\sec^\(2(t) + \sec(t)\tan(t)) \, dt
  2. Integrate each term: Integrate each term separately.\newlineCalculation: ∫sec⁡2(t) dt+∫sec⁡(t)tan⁡(t) dt\int \sec^2(t) \, dt + \int \sec(t)\tan(t) \, dt
  3. Find integrals: The integral of sec⁡2(t)\sec^2(t) is tan⁡(t)\tan(t), and the integral of sec⁡(t)tan⁡(t)\sec(t)\tan(t) is sec⁡(t)\sec(t).\newlineCalculation: tan⁡(t)+sec⁡(t)+C\tan(t) + \sec(t) + C
  4. Combine results: Combine the results to get the final answer.\newlineCalculation: tan⁡(t)+sec⁡(t)+C\tan(t) + \sec(t) + C

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