Let α > 0 . If ∫ 0 α x x + α - x d x = 16 + 20 2 15 then α is equal to :

Step-by-step solution

Given, ∫ 0 α x x + α - x d x = 16 + 20 2 15 . . . . ( 1 ) Rationalise the denominator in L.H.S we get, ∫ 0 α x ( α + x + x ) α d x = 1 α ∫ 0 α x α + x d x + ∫ 0 α x 3 / 2 d x Put α + x = t 2 ⇒ d x = 2 tdt Limit when x = 0 → t = α ,   x = α → t = 2 α 1 α ∫ α 2 α t 2 - α 2 t 2 dt + 2 α 3 / 2 5 = 1 α 2 t 5 5 - 2 α t 3 3 α 2 α + 2 α 3 / 2 5 = 4 α 2