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Ex 7.11, 2 - Using properties of definite integrals, evaluate - Ex 7.11

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.11, 2 By using the properties of definite integrals, evaluate the integrals : 0﷮ 𝜋﷮2﷯﷮ ﷮ sin﷮𝑥﷯﷯﷮ ﷮ sin﷮𝑥﷯﷯ + ﷮ cos﷮𝑥﷯﷯ ﷯ 𝑑𝑥﷯ Let I= 0﷮ 𝜋﷮2﷯﷮ ﷮ sin﷮𝑥﷯﷯﷮ ﷮ sin﷮𝑥﷯﷯ + ﷮ cos﷮𝑥﷯﷯ ﷯ 𝑑𝑥﷯ I= 0﷮ 𝜋﷮2﷯﷮ ﷮ sin ﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯﷮ ﷮ sin ﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯ + ﷮ cos ﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯ ﷯ 𝑑𝑥﷯ ∴ I= 0﷮ 𝜋﷮2﷯﷮ ﷮ 𝑐𝑜𝑠 ﷮𝑥﷯﷯﷮ ﷮ 𝑐𝑜𝑠 ﷮𝑥﷯﷯ + ﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ ﷯ 𝑑𝑥﷯ Adding (1) and (2) i.e. (1) + (2) I+I= 0﷮ 𝜋﷮2﷯﷮ ﷮ sin ﷮𝑥﷯﷯﷮ ﷮ sin ﷮𝑥﷯﷯ + ﷮ cos ﷮𝑥﷯﷯ ﷯ 𝑑𝑥﷯+ 0﷮ 𝜋﷮2﷯﷮ ﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯﷮ ﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯ + ﷮ 𝑠𝑖𝑛﷮𝑥﷯﷯ ﷯ 𝑑𝑥﷯ 2I= 0﷮ 𝜋﷮2﷯﷮ ﷮ sin ﷮𝑥﷯﷯ + ﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯﷮ ﷮ sin ﷮𝑥﷯﷯ + ﷮ 𝑐𝑜𝑠﷮𝑥﷯﷯﷯ ﷯ 𝑑𝑥﷯ 2I= 0﷮ 𝜋﷮2﷯﷮ 𝑑𝑥﷯ I= 1﷮2﷯ 0﷮ 𝜋﷮2﷯﷮ 𝑑𝑥﷯ I= 1﷮2﷯ 𝑥﷯﷮0﷮ 𝜋﷮2﷯﷯ I= 1﷮2﷯ 𝜋﷮2﷯−0﷯ ∴ I= 𝜋﷮4﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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