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Misc 42 - Intgeral cos 2x / (sin x + cos x) dx is equal - Integration using trigo identities - 2x formulae

Misc 42 - Chapter 7 Class 12 Integrals - Part 2

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Transcript

Misc 42 Choose the correct answer ﷐﷮﷮﷐﷐cos﷮2𝑥﷯﷮﷐﷐﷐sin﷮𝑥﷯ + ﷐cos﷮𝑥﷯﷯﷮2﷯ ﷯﷯ 𝑑𝑥 is equal to (A) ﷐−1﷮﷐sin﷮𝑥﷯ + ﷐cos﷮𝑥﷯﷯+𝐶 (B) ﷐log﷮﷐﷐sin﷮𝑥﷯+﷐cos﷮𝑥﷯﷯﷯+𝐶 (C) ﷐log﷮﷐﷐sin﷮𝑥﷯−﷐cos﷮𝑥﷯﷯﷯+𝐶 (D) ﷐1﷮﷐﷐﷐sin﷮𝑥﷯ + ﷐cos﷮𝑥﷯﷯﷮2﷯﷯ ﷐﷮﷮﷐﷐cos﷮2𝑥﷯﷮﷐﷐﷐cos﷮𝑥﷯ + ﷐sin﷮𝑥﷯﷯﷮2﷯﷯﷯ =﷐﷮﷮﷐﷐﷐cos﷮2﷯﷮𝑥﷯ − ﷐﷐sin﷮2﷯﷮𝑥﷯﷮﷐﷐﷐cos﷮𝑥﷯ + ﷐sin﷮𝑥﷯﷯﷮2﷯﷯﷯ 𝑑𝑥 =﷐﷮﷮﷐﷐﷐cos﷮𝑥﷯ − ﷐sin﷮𝑥﷯﷯﷐﷐cos﷮𝑥﷯ + ﷐sin﷮𝑥﷯﷯﷮﷐﷐﷐cos﷮𝑥﷯ + ﷐sin﷮𝑥﷯﷯﷮2﷯﷯﷯ 𝑑𝑥 =﷐﷮﷮﷐﷐cos﷮𝑥﷯ − ﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯ + ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑥 Let ﷐cos﷮𝑥﷯+﷐sin﷮𝑥﷯=𝑡 Diff w.r.t. x −﷐sin﷮𝑥﷯+﷐cos﷮𝑥﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥=﷐1﷮﷐﷐cos﷮𝑥﷯ − ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑡 Hence, our equation becomes =﷐﷮﷮﷐﷐﷐cos﷮𝑥﷯ − ﷐sin﷮𝑥﷯﷯﷮𝑑𝑡﷯﷯ ×﷐𝑑𝑡﷮﷐cos﷮𝑥﷯ − ﷐sin﷮𝑥﷯﷯ =﷐﷮﷮﷐1﷮𝑡﷯﷯ 𝑑𝑡 =﷐log﷮𝑡﷯+𝐶 Putting value of 𝑡=𝑐𝑜𝑠⁡𝑥+𝑠𝑖𝑛⁡𝑥 =﷐log﷮﷐﷐cos﷮𝑥﷯+﷐sin﷮𝑥﷯﷯﷯+𝐶 Hence correct answer is B.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.