Consider I 1 = ∫ 10 20 ln ⁡ x ln ⁡ x + ln ⁡ 30 - x d x and I 2 = ∫ 20 30 ln ⁡ x ln ⁡ x + ln ⁡ 50 - x d x . Then, the value of I 1 I 2 is

Step-by-step solution

Applying a + b - x in I 1 , and adding, we get, 2 I 1 = ∫ 10 20 ln ⁡ x + ln ⁡ 30 - x ln ⁡ x + ln ⁡ 30 - x d x 2 I 1 = ∫ 10 20 1 d x = x 10 20 = 20 - 10 ⇒ I 1 = 5 Similarly, 2 I 2 = x 20 30 = 30 - 20 ⇒ I 2 = 5 ∴ I 1 I 2 = 1