If I = ∫ ln ⁡ x 5 x 2 + x 2 ln ⁡ x 3 d x = k ln ⁡ x 3 + 1 ln ⁡ x 3 - 2 + c (where c is the constant of integration), then 9 k is equal to

Step-by-step solution

Let 1 + ln ⁡ x 3 = t 2 ⇒ I = ∫ t 2 - 1 2 t d t 3 t ⇒ I = 2 3 ∫ t 2 - 1 d t = 2 t 3 9 - 2 3 t + c = 2 9 t t 2 - 3 + c = 2 9 1 + ln ⁡ x 3 ln ⁡ x 3 - 2 + c ∴ k = 2 9 ⇒ 9 k = 2