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Ex 7.3, 3 - Integrate cos 2x cos 4x cos 6x - Ex 7.3

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Transcript

Ex 7.3, 3 cos 2x cos 4x cos 6x We know that 2 cos A cos B= cos﷮ 𝐴+𝐵﷯﷯+ cos﷮ 𝐴−𝐵﷯﷯﷯ Replace A by 2𝑥 & B by 4𝑥 2 cos 2𝑥 cos 4𝑥= cos﷮ 2𝑥+4𝑥﷯﷯+ cos﷮ 2𝑥−4𝑥﷯﷯ 2 cos 2𝑥 cos 4𝑥﷯= cos﷮ 6𝑥﷯﷯+ cos﷮ −2𝑥﷯﷯ 2 cos 2𝑥 cos 4𝑥= cos﷮6𝑥﷯+ cos﷮2𝑥﷯ cos 2𝑥 cos 4𝑥= 1﷮2﷯ cos﷮6𝑥﷯+ cos﷮2𝑥﷯﷯ ﷮﷮ cos﷮2𝑥﷯ cos﷮4𝑥﷯ cos﷮6𝑥﷯﷯﷯𝑑𝑥= ﷮﷮ 1﷮2﷯ cos﷮6𝑥﷯+ cos﷮2𝑥﷯﷯ cos﷮6𝑥﷯﷯﷯𝑑𝑥 = 1﷮2﷯ ﷮﷮ cos﷮6𝑥﷯+ cos﷮2𝑥﷯﷯ cos﷮6𝑥﷯﷯𝑑𝑥 = 1﷮2﷯ ﷮﷮ cos﷮6𝑥﷯﷯﷮2﷯+ cos﷮2𝑥﷯. cos﷮6𝑥﷯﷯﷯𝑑𝑥 = 1﷮2﷯ ﷮﷮ cos﷮6𝑥﷯﷯﷮2﷯ 𝑑𝑥+﷯ ﷮﷮ cos﷮2𝑥﷯. cos﷮6𝑥﷯﷯𝑑𝑥﷯ Now, ﷮﷮ cos﷮2𝑥﷯ cos﷮4𝑥﷯ cos﷮6𝑥﷯﷯﷯𝑑𝑥= 1﷮2﷯ ﷮﷮ cos﷮2﷯﷮6𝑥﷯﷯𝑑𝑥+ ﷮﷮ 𝑐𝑜𝑠﷮2𝑥﷯﷯ cos﷮6𝑥﷯𝑑𝑥﷯ = 1﷮2﷯ 1﷮2﷯ ﷮﷮ 𝑐𝑜𝑠﷮12𝑥+1﷯﷯﷯𝑑𝑥+ 1﷮2﷯ ﷮﷮ 𝑐𝑜𝑠﷮8𝑥﷯+ cos﷮4𝑥﷯﷯﷯𝑑𝑥﷯ = 1﷮4﷯ ﷮﷮ 𝑐𝑜𝑠﷮12𝑥+1﷯﷯﷯𝑑𝑥+ ﷮﷮( 𝑐𝑜𝑠﷮8𝑥﷯﷯+ cos﷮4𝑥﷯)𝑑𝑥﷯ = 1﷮4﷯ ﷮﷮ 𝑐𝑜𝑠﷮12𝑥﷯﷯𝑑𝑥+ ﷮﷮1﷯𝑑𝑥+ ﷮﷮ cos﷮8𝑥﷯﷯𝑑𝑥+ ﷮﷮ cos﷮4𝑥﷯﷯𝑑𝑥﷯ = 𝟏﷮𝟒﷯ 𝐬𝐢𝐧﷮𝟏𝟐𝒙﷯﷮𝟏𝟐﷯ +𝒙+ 𝒔𝒊𝒏﷮𝟖𝒙﷯﷮𝟖﷯ + 𝒔𝒊𝒏﷮𝟒𝒙﷯﷮𝟒﷯﷯+𝑪

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