Ex 7.3, 19 - Integrate 1 / sin x. cos3 x - Chapter 7 - Integration using trigo identities - sin^2,cos^2 etc formulae

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.3, 19 1﷮ sin﷮𝑥﷯ . cos﷮3﷯﷮𝑥﷯﷯ ﷮﷮ 1﷮ sin﷮𝑥﷯ . cos﷮3﷯﷮𝑥﷯﷯﷯ 𝑑𝑥= ﷮﷮ sin﷮2﷯﷮𝑥﷯ + cos﷮2﷯﷮𝑥﷯﷮ sin﷮𝑥﷯ . cos﷮3﷯﷮𝑥﷯﷯﷯ 𝑑𝑥 = ﷮﷮ sin﷮2﷯﷮𝑥﷯﷮ sin﷮𝑥﷯ . cos﷮3﷯﷮𝑥﷯﷯+ cos﷮2﷯﷮𝑥﷯﷮ sin﷮𝑥﷯ . cos﷮3﷯﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 𝑠𝑖𝑛﷮𝑥﷯﷮ cos﷮3﷯﷮𝑥﷯﷯+ 𝑐𝑜𝑠﷮𝑥﷯﷮ sin﷮𝑥﷯ . cos﷮2﷯﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 𝑠𝑖𝑛﷮𝑥﷯﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯× 1﷮ cos﷮2﷯﷮𝑥﷯﷯+ 𝑐𝑜𝑠﷮𝑥﷯﷮ sin﷮𝑥﷯﷯× 1﷮ cos﷮2﷯﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ cos﷮2﷯﷮𝑥﷯﷯ 𝑠𝑖𝑛﷮𝑥﷯﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯+ 𝑐𝑜𝑠﷮𝑥﷯﷮ sin﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ sec﷮2﷯﷮𝑥﷯ tan﷮𝑥﷯+ cot﷮𝑥﷯﷯﷯ 𝑑𝑥 = ﷮﷮ sec﷮2﷯﷮𝑥﷯ tan﷮𝑥﷯+ 1﷮ tan﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ tan﷮𝑥﷯+ 1﷮ tan﷮𝑥﷯﷯﷯. sec﷮2﷯﷮𝑥﷯﷯ 𝑑𝑥 Putting tan⁡x=t Differentiating w.r.t.x sec﷮2﷯﷮𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 1﷮ sec﷮2﷯﷮𝑥﷯﷯ 𝑑𝑡 Thus, our equation becomes = ﷮﷮ 𝑡+ 1﷮𝑡﷯﷯ sec﷮2﷯﷮𝑥﷯× 1﷮ sec﷮2﷯﷮𝑥﷯﷯﷯ 𝑑𝑡 = ﷮﷮ 𝑡+ 1﷮𝑡﷯﷯﷯ 𝑑𝑡 = ﷮﷮𝑡﷯ 𝑑𝑡+ ﷮﷮ 1﷮𝑡﷯﷯ 𝑑𝑡 = 𝑡﷮2﷯﷮2﷯ + log﷮ 𝑡﷯﷯+𝐶 Putting value of 𝑡=𝑡𝑎𝑛⁡𝑥 = 𝒍𝒐𝒈﷮ 𝒕𝒂𝒏﷮𝒙﷯﷯﷯+ 𝒕𝒂𝒏﷮𝟐﷯ 𝒙﷮𝟐﷯ +𝑪

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