For real numbers α , β , γ and δ , if ∫ x 2 - 1 + tan - 1 x 2 + 1 x x 4 + 3 x 2 + 1 tan - 1 x 2 + 1 x d x = α log e tan - 1 x 2 + 1 x + β tan - 1 γ x 2 - 1 x + δ tan - 1 x 2 + 1 x + C where C is an arbitrary constant, then the value of 10 ( α + β γ + δ ) is equal to ______ .

Step-by-step solution

∫ x 2 - 1 d x x 4 + 3 x 2 + 1 tan - 1 x + 1 x + ∫ d x x 4 + 3 x 2 + 1 ∫ 1 - 1 x 2 d x x + 1 x 2 + 1 tan - 1 x + 1 x + 1 2 ∫ x 2 + 1 - x 2 - 1 d x x 4 + 3 x 2 + 1 Put tan - 1 x + 1 x = t ∫ d t t + 1 2 ∫ 1 + 1 x 2 d x x - 1 x 2 + 5 - 1 2 ∫ 1 - 1 x 2 d x x + 1 x 2 + 1 Put x - 1 x = y , x + 1 x = z log e t + 1 2 ∫ d y y 2 + 5 - 1 2 ∫ d z z 2 + 1 = log e tan - 1 x + 1 x + 1 2 5 tan - 1 x 2 - 1 5 x - 1 2 tan - 1 x 2 + 1 x + C α = 1 , β = 1 2 5 , γ = 1 5