Ex 7.11, 4 - Evaluate integral cos5 x dx / sin5 x + cos5 x - Ex 7.11

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

Transcript

Ex 7.11, 4 By using the properties of definite integrals, evaluate the integrals : 0﷮ 𝜋﷮2﷯﷮ cos﷮5﷯﷮𝑥﷯ 𝑑𝑥﷮ sin﷮5﷯﷮𝑥﷯ + cos﷮5﷯﷮𝑥﷯﷯﷯ Let I= 0﷮ 𝜋﷮2﷯﷮ cos﷮5﷯﷮𝑥﷯﷮ sin﷮5﷯﷮𝑥﷯ + cos﷮5﷯﷮𝑥﷯﷯ 𝑑𝑥﷯ I= 0﷮ 𝜋﷮2﷯﷮ cos﷮5﷯ 𝜋﷮2﷯ − 𝑥﷯﷮ 𝑠𝑖𝑛﷮5﷯ 𝜋﷮2﷯ − 𝑥﷯ + 𝑐𝑜𝑠﷮5﷯ 𝜋﷮2﷯ − 𝑥﷯﷯ 𝑑𝑥﷯ ∴ I = 0﷮ 𝜋﷮2﷯﷮ sin﷮5﷯﷮𝑥﷯﷮ cos﷮5﷯﷮𝑥﷯ + sin﷮5﷯﷮𝑥﷯﷯ 𝑑𝑥﷯ Adding (1) and (2) i.e. (1) + (2) I+I= 𝑐𝑜𝑠﷮5﷯ 𝑥﷮ 𝑠𝑖𝑛﷮5﷯ 𝑥 + 𝑐𝑜𝑠﷮5﷯ 𝑥﷯ 𝑑𝑥+ 0﷮ 𝜋﷮2﷯﷮ sin﷮5﷯﷮𝑥﷯﷮ cos﷮5﷯﷮𝑥﷯ + sin﷮5﷯﷮𝑥﷯﷯𝑑𝑥﷯ 2I= 0﷮ 𝜋﷮2﷯﷮ 𝑐𝑜𝑠﷮5﷯ 𝑥 + 𝑠𝑖𝑛﷮5﷯ 𝑥﷮ 𝑐𝑜𝑠﷮5﷯ 𝑥 + 𝑠𝑖𝑛﷮5﷯ 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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