If ₀^ 8 2 ^2 x x (1 + 3^ x ) (1 + ^4 x ) dx = - _e ( + 12 2 ) , where , are integers, then the value of + is __________

Step-by-step solution

Let I = ₀^ 8 2 ^2 x x (1 + 3^ x ) (1 + ^4 x ) dx (i) Using the property ₀^a f(x) dx = ₀^a f(a-x) dx , we replace x with - x : I = ₀^ 8 2 (- x)^2 x (1 + 3^ - x ) (1 + (- x)^4 ) dx I = ₀^ 8 2 ^2 x x 3^ x (1 + 3^ x ) (1 + ^4 x ) dx (ii) Adding (i) and (ii) : 2I = ₀^ 8 2 ^2 x x 1 + ^4 x dx I = ₀^ /2 8 2 ^2 x x 1 + ^4 x dx (using ₀^ 2a f(x) dx = 2 ₀^a f(x) dx for f(2a-x)=f(x) ) Let x = t - x dx = dt . When x = 0, t = 1 and when x = /2, t = 0 . I = ₀¹ 8 2 t^2 1 + t^4 dt = 4 2 ₀¹ 2t^2 t^4 + 1 dt I = 4 2 ₀¹ ( 1 + 1 t^2 t^2