If ₀¹ 4 ⁻¹ (1-2 x+4 x² ) d x= atan ⁻¹(2)- blog _ e (5) , where a , b N , then (2 a + b ) is equal to _ _ _ _ .

Step-by-step solution

Let I= ₀^1 ⁻¹ (1-2 x+4 x^2 ) d x I= ₀^1 ( ⁻¹(2 x-1)- ⁻¹(2 x) ) d x Applying king I= ₀^1 (- ⁻¹(2 x-1)+ ⁻¹(2 x-2) ) d x From (1) and (2) 2 I= ₀^1 ( ⁻¹(2 x-2)- ⁻¹(2 x) ) d x = ₀^1 ⁻¹(2 x-2) d x- ₀^1 ⁻¹(2 x) d x Applying King = ₀^1 ⁻¹(-2 x) d x- ₀^1 ⁻¹(2 x) d x = ₀^1 ( - ⁻¹(2 x) ) d x- ₀^1 ⁻¹(2 x) d x = ₀^1 ( -2 ⁻¹(2 x) ) d x = -2 ₀^1 ( ⁻¹ 2 x ) 1 d x By parts = -2 [ (x ⁻¹ 2 x )₀^1+ ₀^1 2 x 1+4 x^2 d x ] Let 1+4 x ^2= t 8 xdx = dt = -2 [ ⁻¹ 2+ 1 4 ₁^5 dt t ] = -2 ⁻¹ 2- 1 2 n 5 2 I =2 ⁻¹ 2- 1 2 n 5 4 I =4 ⁻¹ 2- n 5 2 a