Misc 7 - Integrate sin x / sin (x - a) - Chapter 7 Class 12 - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
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Misc 7 Integrate the function sin﷮𝑥﷯﷮ sin﷮ 𝑥 − 𝑎﷯﷯﷯ Let I = ﷮﷮ sin﷮𝑥﷯﷮ sin﷮ 𝑥 − 𝑎﷯﷯﷯﷯ 𝑑𝑥 Put t = 𝑥 − 𝑎 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑡﷮𝑑𝑥﷯ = 𝑑 𝑥 − 𝑎﷯﷮𝑑𝑥﷯ 𝑑𝑡﷮𝑑𝑥﷯ = 1 𝑑𝑥 = 𝑑𝑡 Therefore ﷮﷮ sin ﷮ 𝑡 + 𝑎﷯﷯﷮ sin﷮𝑡﷯﷯﷯ 𝑑𝑡 = ﷮﷮ sin﷮𝑡﷯ cos﷮𝑎﷯ + cos﷮𝑡﷯ sin﷮𝑎﷯﷮ sin﷮𝑡﷯﷯﷯ 𝑑𝑡 = ﷮﷮ sin﷮𝑡﷯ cos﷮𝑎﷯﷮ sin﷮𝑡﷯﷯ + cos﷮𝑡﷯ sin﷮𝑎﷯﷮ sin﷮𝑡﷯﷯﷯﷯𝑑𝑡 = ﷮﷮ cos﷮𝑎﷯﷯𝑑𝑡 + ﷮﷮ cot﷮𝑡﷯﷯ sin﷮𝑎﷯ 𝑑𝑡 = cos﷮𝑎﷯ ﷮﷮𝑑𝑡﷯ + sin﷮𝑎﷯ ﷮﷮ cot﷮𝑡﷯﷯ 𝑑𝑡 = t cos a + sin a log sin﷮𝑡﷯﷯ + C Putting back t = x – a = (x – a) cos a + sin a sin﷮(𝑥+𝑎)﷯﷯ + C = sin a log sin﷮(𝑥−𝑎)﷯﷯ + x cos x − a cos a + C = sin a log 𝐬𝐢𝐧﷮(𝒙−𝒂)﷯﷯ + x cos x + 𝐂﷮𝟏﷯

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