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∫tan⁡2(x)+1+tan⁡(x)sec⁡(x) dx\int \tan^{2}(x) + 1 + \tan(x) \sec(x) \, dx=

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Q. ∫tan⁡2(x)+1+tan⁡(x)sec⁡(x) dx\int \tan^{2}(x) + 1 + \tan(x) \sec(x) \, dx=
  1. Simplify using trigonometric identities: Simplify the integrand using trigonometric identities.\newlineWe know that tan⁡2(x)+1=sec⁡2(x)\tan^2(x) + 1 = \sec^2(x) and that tan⁡(x)sec⁡(x)\tan(x) \sec(x) is the derivative of sec⁡(x)\sec(x). So, the integral becomes:\newline∫(tan⁡2(x)+1+tan⁡(x)sec⁡(x)) dx=∫(sec⁡2(x)+tan⁡(x)sec⁡(x)) dx\int(\tan^2(x) + 1 + \tan(x) \sec(x))\,dx = \int(\sec^2(x) + \tan(x) \sec(x))\,dx
  2. Split into two integrals: Split the integral into two separate integrals.\newline\int(\sec^\(2(x) + \tan(x) \sec(x))\,dx = \int\sec^22(x)\,dx + \int\tan(x) \sec(x)\,dx
  3. Integrate sec⁡2(x)\sec^2(x): Integrate ∫sec⁡2(x) dx\int \sec^2(x)\,dx. The integral of sec⁡2(x)\sec^2(x) with respect to xx is tan⁡(x)\tan(x), so we have: ∫sec⁡2(x) dx=tan⁡(x)\int \sec^2(x)\,dx = \tan(x)
  4. Integrate tan⁡(x)sec⁡(x)\tan(x) \sec(x): Integrate ∫tan⁡(x)sec⁡(x) dx\int \tan(x) \sec(x)\,dx. Since tan⁡(x)sec⁡(x)\tan(x) \sec(x) is the derivative of sec⁡(x)\sec(x), the integral of tan⁡(x)sec⁡(x)\tan(x) \sec(x) with respect to xx is sec⁡(x)\sec(x), so we have: ∫tan⁡(x)sec⁡(x) dx=sec⁡(x)\int \tan(x) \sec(x)\,dx = \sec(x)
  5. Combine results: Combine the results from Step 33 and Step 44.\newlineThe integral of the original function is the sum of the integrals found in Step 33 and Step 44, so we have:\newline∫(tan⁡2(x)+1+tan⁡(x)sec⁡(x)) dx=tan⁡(x)+sec⁡(x)+C\int(\tan^2(x) + 1 + \tan(x) \sec(x))\,dx = \tan(x) + \sec(x) + C, where CC is the constant of integration.

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