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Ex 7.11, 15 - Evaluate sin x - cos x / 1 + sin x cos x dx - Definate Integration by properties - P4

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.11, 15 By using the properties of definite integrals, evaluate the integrals : 0﷮ 𝜋﷮2﷯﷮ sin﷮𝑥﷯ − cos﷮𝑥﷯﷮1 + sin﷮𝑥﷯ cos﷮𝑥﷯﷯﷯ 𝑑𝑥 Let I= 0﷮ 𝜋﷮2﷯﷮ sin﷮𝑥 − cos﷮𝑥﷯﷯﷮1 + sin﷮𝑥 cos﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ ∴ I= 0﷮ 𝜋﷮2﷯﷮ 𝑠𝑖𝑛 𝜋﷮2﷯ − 𝑥﷯ − 𝑐𝑜𝑠 𝜋﷮2﷯ − 𝑥﷯﷮1 + 𝑠𝑖𝑛 𝜋﷮2﷯ − 𝑥﷯ 𝑐𝑜𝑠 𝜋﷮2﷯ − 𝑥﷯﷯ 𝑑𝑥﷯ I= 0﷮ 𝜋﷮2﷯﷮ cos﷮𝑥﷯ − sin﷮𝑥﷯﷮ 1 + cos﷮𝑥﷯ sin﷮𝑥 ﷯﷯ 𝑑𝑥﷯ Adding (1) and (2)i.e. (1) + (2) I+I= 0﷮ 𝜋﷮2﷯﷮ sin﷮𝑥 − cos﷮𝑥﷯﷯﷮1 + sin﷮𝑥 cos﷮𝑥﷯﷯﷯ 𝑑𝑥+﷯ 0﷮ 𝜋﷮2﷯﷮ cos﷮𝑥 − sin﷮𝑥﷯﷯﷮1 + sin﷮𝑥 cos﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ 2I= 0﷮ 𝜋﷮2﷯﷮ sin﷮𝑥 − cos﷮𝑥 + cos﷮𝑥 − sin﷮𝑥﷯﷯﷯﷯﷮1 + sin﷮𝑥 cos﷮𝑥﷯﷯﷯ ﷯𝑑𝑥 2I= 0﷮ 𝜋﷮2﷯﷮ 0﷮1 + sin﷮𝑥 cos﷮𝑥﷯﷯﷯ ﷯𝑑𝑥 2I=0 ∴ 𝐈=𝟎

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