If _ e^2 ^ e^a 1 x ( ( _e x)^3 ( _e x)^3 + (12 - _e x)^3 ) dx = 4 , where a > 2 , then the value of a is

Step-by-step solution

Let I = _ e^2 ^ e^a 1 x ( ( _e x)^3 ( _e x)^3 + (12 - _e x)^3 ) dx Substitute t = _e x dt = 1 x dx . When x = e^2 , t = 2 . When x = e^a , t = a . I = ₂^a t^3 t^3 + (12 - t)^3 dt ... (i) Using the property _p^q f(t) dt = _p^q f(p+q-t) dt , we get: I = ₂^a (2+a-t)^3 (2+a-t)^3 + (12 - (2+a-t))^3 dt For the denominator to remain the same as in (i), we must have 2+a = 12 , which gives a = 10 . Let us verify if a = 10 satisfies the given equation: I = ₂¹⁰ t^3 t^3 + (12 - t)^3 dt Applying the property again: I = ₂¹⁰ (12-