Misc 21 - Integarate 2 + sin 2x / 1 + cos 2x ex  - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 21 Integrate the function 2 + sin﷮2𝑥﷯﷮1 + cos﷮2𝑥﷯﷯ 𝑒﷮𝑥﷯ We can write integral as 2 + sin﷮2𝑥﷯﷮1 + cos﷮2𝑥﷯﷯﷯ 𝑒﷮𝑥﷯ = 2 + sin﷮2𝑥﷯﷮1 + 2 cos﷮2﷯﷮𝑥 − 1﷯﷯﷯﷯ 𝑒﷮𝑥﷯ = 2 + sin﷮2𝑥﷯﷮2 cos﷮2﷯﷮𝑥﷯﷯﷯ 𝑒﷮𝑥﷯ = 2 + 2 cos﷮𝑥 sin﷮𝑥﷯﷯﷮2 cos﷮2﷯﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = 2 1 + cos﷮𝑥 sin﷮𝑥﷯﷯﷯﷮2 cos﷮2﷯﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = 1 + cos﷮𝑥 sin﷮𝑥﷯﷯﷮ cos﷮2﷯﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = 1﷮ cos﷮2﷯﷮𝑥﷯﷯+ cos﷮𝑥 sin﷮𝑥﷯﷯﷮ cos﷮2﷯﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = sec﷮2﷯﷮𝑥+ cos﷮𝑥 sin﷮𝑥﷯﷯﷮ cos﷮𝑥 cos﷮𝑥﷯﷯﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = sec﷮2﷯﷮𝑥+ tan﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ = tan﷮𝑥+ sec﷮2﷯﷮𝑥﷯﷯﷯﷮ 𝑒﷮𝑥﷯﷯ Let f(x) = tan x f’(x) = 𝑠𝑒𝑐﷮2﷯x Integrate w.r.t.𝑥 ﷮﷮ 2+ sin﷮2𝑥﷯﷮1+ cos﷮2𝑥﷯﷯﷯ 𝑒﷮𝑥﷯ 𝑑﷮𝑥﷯= ﷮﷮ 𝑒﷮𝑥﷯ tan﷮𝑥+ sec﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷯﷯ = 𝒆﷮𝒙﷯ 𝒕𝒂𝒏﷮𝒙﷯+𝐂

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