Ex 7.3, 15 - Integrate tan3 2x sec 2x - Class 12 NCERT - Ex 7.3

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.3, 15 tan3 2𝑥 sec⁡2𝑥 Let I= ﷮﷮𝑡𝑎𝑛3 2𝑥 sec﷮2𝑥﷯𝑑𝑥﷯ = ﷮﷮ tan﷮2﷯﷮2𝑥﷯. tan﷮2𝑥﷯. sec﷮2𝑥﷯﷯﷯𝑑𝑥 = ﷮﷮ sec﷮2﷯﷮2𝑥−1﷯﷯ 𝑡𝑎𝑛﷮2𝑥﷯. sec﷮2𝑥﷯﷯𝑑𝑥 Putting sec⁡2𝑥=𝑡 Differentiating w.r.t.x 𝑑 sec﷮2﷯𝑥﷯﷮𝑑𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 2﷮sec﷯2𝑥 tan﷮2𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮ 2﷮sec ﷯2𝑥 tan﷮2𝑥﷯﷯ Putting values of sec 2x and dx in equation I = ﷮﷮ sec﷮2﷯﷮2𝑥−1﷯﷯ 𝑡𝑎𝑛﷮2𝑥﷯. sec﷮2𝑥﷯﷯𝑑𝑥 = ﷮﷮ 𝑡﷮2﷯−1﷯ 𝑡𝑎𝑛﷮2𝑥﷯. sec﷮2𝑥﷯﷯× 𝑑𝑡﷮ 2﷮sec ﷯2𝑥 tan﷮2𝑥﷯﷯ = ﷮﷮ 𝑡﷮2﷯−1﷯﷯× 𝑑𝑡﷮2﷯ = 1﷮2﷯ ﷮﷮ 𝑡﷮2﷯−1﷯﷯𝑑𝑡 = 1﷮2﷯ ﷮﷮ 𝑡﷮2﷯﷯𝑑𝑡− ﷮﷮1﷯𝑑𝑥﷯ = 1﷮2﷯ 𝑡﷮2 + 1﷯﷮2 + 1﷯ −𝑡+𝐶﷯ = 1﷮2﷯ 𝑡﷮3﷯﷮3﷯ −𝑡+𝐶﷯ Putting value of 𝑡=𝑠𝑒𝑐⁡2𝑥 = 1﷮2﷯ sec﷮3﷯﷮2𝑥 ﷯﷮3﷯ − sec﷮2𝑥﷯﷯+𝐶﷯ = 𝒔𝒆𝒄﷮𝟑﷯﷮𝟐𝒙 ﷯﷮𝟔﷯ − 𝒔𝒆𝒄﷮𝟐𝒙﷯﷮𝟐﷯+𝑪

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.
  • Shubham Rajpurohit's image
    Shubham Rajpurohit
    Feb. 19, 2018, 11:33 a.m.
    Sir question no 15 mein ex 7.3 mein Apne sec2x Ko T put Kiya agar hum pure hi secx-1 to t put krde to -1 to constant hai vi to 0 hojayega aur humara answer bhi kuch aur aaraha hai
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