Ex 7.3, 7 - Integrate sin 4x sin 8x - Chapter 7 Class 12 - Ex 7.3

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.3, 7 𝑠𝑖𝑛 4𝑥 sin⁡8𝑥 ﷮﷮ sin﷮4𝑥﷯sin⁡8𝑥﷯ 𝑑𝑥 We know that 2 𝑠𝑖𝑛﷮𝐴﷯ 𝑠𝑖𝑛﷮𝐵﷯=− 𝑐𝑜𝑠﷮ 𝐴+𝐵﷯﷯+ 𝑐𝑜𝑠﷮ 𝐴−𝐵﷯﷯ 𝑠𝑖𝑛﷮𝐴﷯ 𝑠𝑖𝑛﷮𝐵﷯= 1﷮2﷯ − 𝑐𝑜𝑠﷮ 𝐴+𝐵﷯﷯+ 𝑐𝑜𝑠﷮ 𝐴−𝐵﷯﷯﷯ Replace A by 4𝑥 & B by 8𝑥 sin﷮4𝑥﷯ sin﷮8𝑥﷯= 1﷮2﷯ − cos﷮ 4𝑥+8𝑥﷯﷯+ cos﷮ 4𝑥−8𝑥﷯﷯﷯ sin﷮4𝑥﷯ sin﷮8𝑥﷯= 1﷮2﷯ − cos﷮12𝑥﷯+ cos﷮ −4𝑥﷯﷯﷯ sin﷮4𝑥﷯ sin﷮8𝑥﷯= 1﷮2﷯ − cos﷮12𝑥﷯+ cos﷮4𝑥﷯﷯ sin﷮4𝑥﷯ sin﷮8𝑥﷯= 1﷮2﷯ cos﷮4𝑥﷯− cos﷮12𝑥﷯﷯ Integrating both sides ﷮﷮ sin﷮4𝑥﷯sin⁡8𝑥﷯ 𝑑𝑥= 1﷮2﷯ ﷮﷮ cos﷮4𝑥﷯﷯𝑑𝑥− ﷮﷮ cos﷮12𝑥﷯﷯𝑑𝑥﷯ = 𝟏﷮𝟐﷯ 𝐬𝐢𝐧﷮𝟒𝒙﷯﷮𝟒﷯ − 𝒔𝒊𝒏﷮𝟏𝟐𝒙﷯﷮𝟏𝟐﷯﷯+𝑪

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