Ex 7.9, 12 - Direct Integrate cos2 x dx from 0 to x/2 - Ex 7.9

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.9, 12 0﷮ 𝑥﷮2﷯﷮𝑐𝑜𝑠2 𝑥 𝑑𝑥﷯ Step 1 :- Let F 𝑥﷯= ﷮﷮𝑐𝑜 𝑠﷮2﷯ 𝑥 𝑑𝑥﷯ = ﷮﷮ cos﷮2𝑥﷯ + 1﷮2﷯﷯ 𝑑𝑥 = 1﷮2﷯ ﷮﷮𝑐𝑜𝑠 2𝑥 𝑑𝑥+ 1﷮2﷯ ﷮﷮𝑑𝑥﷯﷯ = 1﷮2﷯ × 𝑠𝑖𝑛 2𝑥﷮2﷯+ 𝑥﷮2﷯ = 1﷮4﷯ 𝑠𝑖𝑛 2𝑥+ 𝑥﷮2﷯ Hence , F 𝑥﷯= 1﷮4﷯𝑠𝑖𝑛 2𝑥+ 𝑥﷮2﷯ Step 2 :- 0﷮ 𝜋﷮2﷯﷮𝑐𝑜 𝑠﷮2﷯ 𝑥=𝐹 𝜋﷮2﷯﷯−𝐹 0﷯﷯ = 1﷮4﷯𝑠𝑖𝑛 2 × 𝜋﷮2﷯﷯+ 𝜋﷮2﷯﷯﷮2﷯− 1﷮4﷯𝑠𝑖𝑛 2× 0﷮2﷯﷯− 0﷮2﷯ = 1﷮4﷯𝑠𝑖𝑛 𝜋﷯+ 𝜋﷮4﷯− 1﷮4﷯ sin﷮0 −0﷯ = 1﷮4﷯ ×0+ 𝜋﷮4﷯− 1﷮4﷯ ×0−0 = 𝝅﷮𝟒﷯

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