Ex 7.11, 10 - Evaluate (2 log sin x - log sin 2x) dx

Ex 7.11, 10 By using the properties of definite integrals, evaluate the integrals : 0﷮ 𝜋﷮2﷯﷮ 2 log﷮ sin﷮𝑥﷯﷯ − log﷮ sin﷮2𝑥﷯﷯﷯﷯ 𝑑𝑥 Let I1= 0﷮ 𝜋﷮2﷯﷮ 2 log﷮ sin﷮𝑥﷯﷯ − log﷮ sin﷮2𝑥﷯﷯﷯﷯ 𝑑𝑥 I1= 0﷮ 𝜋﷮2﷯﷮ 2 log﷮ sin﷮𝑥﷯﷯−𝑙𝑜𝑔 2 sin﷮𝑥 cos﷮𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 I1= 0﷮ 𝜋﷮2﷯﷮ 2 log﷮ sin﷮𝑥﷯﷯− log﷮2﷯− log﷮ sin﷮𝑥− log﷮ cos﷮𝑥﷯﷯﷯﷯﷯﷯ 𝑑𝑥 I1= 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥﷯﷯−𝑙𝑜𝑔2− log﷮ cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 I1= 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥 𝑑𝑥﷯﷯﷯− 0﷮ 𝜋﷮2﷯﷮ log﷮2𝑑𝑥﷯− 0﷮ 𝜋﷮2﷯﷮ log﷮ cos﷮𝑥 𝑑𝑥﷯﷯﷯﷯ Solving I2 I2= 0﷮ 𝜋﷮2﷯﷮ log﷮ cos﷮𝑥 𝑑𝑥﷯﷯﷯ ∴ I2= 0﷮ 𝜋﷮2﷯﷮ log﷮𝑐𝑜𝑠 𝜋﷮2﷯−𝑥﷯𝑑𝑥﷯﷯ I2= 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥 𝑑𝑥﷯﷯﷯ Put the value of I2 in (1) i.e. I1 ∴ I1= 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥 𝑑𝑥﷯﷯﷯− 0﷮ 𝜋﷮2﷯﷮ log 2﷮𝑑𝑥﷯﷯− 0﷮ 𝜋﷮2﷯﷮ log﷮ cos﷮𝑥 𝑑𝑥﷯﷯﷯ I1= 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥 𝑑𝑥﷯﷯﷯− 0﷮ 𝜋﷮2﷯﷮ log 2﷮𝑑𝑥﷯﷯− 0﷮ 𝜋﷮2﷯﷮ log﷮ sin﷮𝑥 𝑑𝑥﷯﷯﷯ I1= – 0﷮ 𝜋﷮2﷯﷮ log 2﷮𝑑𝑥﷯﷯ I1= – log 2 0﷮ 𝜋﷮2﷯﷮𝑑𝑥﷯ I1= – log 2 𝑥﷯﷮0﷮ 𝜋﷮2﷯﷯ I1= – log 2 𝜋﷮2﷯−0﷯ I1= log﷮ 2﷯﷮−1﷯﷯ 𝜋﷮2﷯﷯ I1= 𝜋﷮2﷯ log 1﷮2﷯﷯