Let I_n = ₀^1 x^2 (1-x^4)^n , dx for n 0 . If I₁₀ = A B I₉ , where A and B are coprime positive integers, then the value of B - A is equal to _______.

Step-by-step solution

Let I_n = ₀^1 x^2 (1-x^4)^n , dx . Using integration by parts, take u = (1-x^4)^n and dv = x^2 , dx . Then du = n(1-x^4)^ n-1 (-4x^3) , dx and v = x^3 3 . I_n = [ (1-x^4)^n x^3 3 ]₀^1 - ₀^1 x^3 3 n(1-x^4)^ n-1 (-4x^3) , dx The first term evaluates to 0 at both limits. I_n = 4n 3 ₀^1 x^6 (1-x^4)^ n-1 , dx Rewrite x^6 as x^2 x^4 = x^2(1 - (1-x^4)) . I_n = 4n 3 ₀^1 x^2 (1 - (1-x^4)) (1-x^4)^ n-1 , dx I_n = 4n 3 ( ₀^1 x^2 (1-x^4)^ n-1 , dx - ₀^1 x^2 (1-x^4)^n , dx ) I_n = 4n 3 (I_ n-1 - I_n) Rearranging the terms: 3 I_