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Example 37 - Evaluate integral cos 6x root 1 + sin 6x dx - Integration by substitution - Trignometric - Normal

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 37 Evaluate ﷮﷮ cos﷮6𝑥 ﷮1+ sin﷮6𝑥﷯﷯﷯﷯ 𝑑𝑥 ﷮﷮ cos﷮6𝑥 ﷮1+ sin﷮6𝑥﷯﷯﷯﷯ 𝑑𝑥 Put 𝑡 = ﷮1+ sin﷮6𝑥﷯﷯ 𝑡﷮2﷯ = 1+ sin﷮6𝑥﷯ Differentiate 𝑤.𝑟.𝑡.𝑥 𝑑 𝑡﷮2﷯﷮𝑑𝑥﷯= 𝑑﷮𝑑𝑥﷯ 1+ sin﷮6𝑥﷯﷯ 2𝑡. 𝑑𝑡﷮𝑑𝑥﷯=6 cos 6 𝑥 2𝑡 𝑑𝑡﷮6 cos﷮6𝑥﷯﷯=𝑑𝑥 Therefore, ﷮﷮ cos﷮6𝑥 ﷮1+ sin﷮6𝑥﷯﷯﷯﷯= ﷮﷮ cos﷮6𝑥 𝑡﷯﷯. 2 𝑡 𝑑𝑡﷮ 6 cos﷮6𝑥﷯﷯ = ﷮﷮ 𝑡﷮2﷯﷮3﷯﷮𝑑𝑡﷯﷯ = 1﷮3﷯ ﷮﷮ 𝑡﷮2﷯﷮𝑑𝑡﷯﷯ = 1﷮3﷯ 𝑡﷮2 + 1﷯﷮2 + 1﷯ + 𝐶 = 1﷮3﷯ 𝑡﷮3﷯﷮3﷯ + 𝐶 = 𝑡﷮3﷯﷮9﷯ + 𝐶 Putting back 𝑡 = ﷮1+ 𝑠𝑖𝑛﷮6𝑥﷯﷯ = ﷮1 + sin﷮6𝑥﷯﷯﷯﷮3﷯﷮9﷯ + 𝐶 = 1 + sin﷮6𝑥﷯﷯﷮ 1﷮2﷯ × 3﷯﷮9﷯ + 𝐶 = 𝟏﷮𝟗﷯ 𝟏 + 𝒔𝒊𝒏﷮𝟔𝒙﷯﷯﷮ 𝟑﷮𝟐﷯﷯+𝑪

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