If ∫ 0 3 15 x 3 1 + x 2 + 1 + x 2 3 d x = α 2 + β 3 , where α , β are integers, then α + β is equal to

Step-by-step solution

Given, ∫ 0 3 15 x 3 1 + x 2 + 1 + x 2 3 d x = α 2 + β 3 , Taking L.H.S we get ∫ 0 3 15 x 3 1 + x 2 + 1 + x 2 3 d x Now put 1 + x 2 = t 2 ⇒ 2 x d x = 2 t d t ⇒ x d t = t d t ∴   ∫ 1 2 15 t 2 - 1 t d t t 2 + t 3 = 15 ∫ 1 2 t t 2 - 1 t 1 + t d t Now put 1 + t = u 2 ⇒ d t = 2 u d u So, the integral will be 15 ∫ 2 3 u 2 - 1 2 - 1 u × 2 u d u = 30 ∫ 2 3 u 4 - 2 u 2 d u = 30 u 5 5 - 2 u 3 3 2 3 = 30 1 5 3 5 - 2 5 - 2 3 3 3 - 2 3 = 30 1 5 9 3 -