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Misc 10 - Integrate sin8 x - cos8 x / 1 - 2 sin2 x cos2 x - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 10 Integrate the function sin﷮8﷯﷮−﷯ cos﷮8﷯﷮𝑥﷯﷮1 −2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯ ﷮﷮ sin﷮8﷯﷮−﷯ cos﷮8﷯﷮𝑥﷯﷮1 −2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮4﷯ 𝑥﷯﷮2﷯﷮− ﷯ cos﷮4﷯ 𝑥﷯﷮2﷯﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮4﷯ 𝑥 + cos﷮4﷯﷮𝑥﷯﷯﷮ sin﷮4﷯﷮𝑥﷯ − cos﷮4﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮4﷯ 𝑥 + cos﷮4﷯﷮𝑥﷯﷯﷮ sin﷮2﷯ 𝑥﷯﷮2﷯ − cos﷮2﷯ 𝑥﷯﷮2﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮4﷯ 𝑥 + cos﷮4﷯﷮𝑥﷯﷯﷮ sin﷮2﷯﷮𝑥﷯ + cos﷮2﷯﷮𝑥﷯﷯﷯ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮4﷯ 𝑥 + cos﷮4﷯﷮𝑥﷯﷯ (1)﷮ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ Adding & Subtracting 2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯ = ﷮﷮ sin﷮4﷯ 𝑥 + cos﷮4﷯﷮𝑥﷯ + 2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯ − 2 sin﷮2﷯﷮ cos﷮2﷯﷮𝑥﷯﷯﷯﷮ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮2﷯﷮𝑥﷯﷯﷮2﷯+ cos﷮2﷯﷮𝑥﷯﷯﷮2﷯ + 2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯﷯−2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯﷯﷮ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮2﷯﷮𝑥﷯ + cos﷮2﷯﷮𝑥﷯﷯﷮2﷯ − 2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯﷯ ﷮ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ 1 − 2 sin﷮2﷯﷮𝑥﷯ cos﷮2﷯﷮𝑥﷯﷯ ﷮ sin﷮2﷯﷮𝑥﷯ − cos﷮2﷯﷮𝑥﷯﷯﷯𝑑𝑥﷮1 − 2 sin﷮2﷯﷮𝑥 cos﷮2﷯﷮𝑥﷯ ﷯﷯﷯ = ﷮﷮ sin﷮2﷯﷮𝑥﷯− cos﷮2﷯﷮𝑥﷯﷯﷯ 𝑑𝑥 =− ﷮﷮ cos﷮2﷯﷮𝑥﷯− sin﷮2﷯﷮𝑥﷯﷯﷯ 𝑑𝑥 =− ﷮﷮ cos﷮2𝑥﷯﷯. 𝑑𝑥 =− 𝟏﷮𝟐﷯ 𝐬𝐢𝐧﷮𝟐𝒙﷯+𝑪

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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