If I_n = ₁^e ( _e x)^n ,dx where n Z^+ and I_m + mI₂₀₂₆ = e , then m =

Step-by-step solution

I_n = ₁^e ( _e x)^n , dx Using integration by parts, taking 1 as the second function: I_n = [ x ( _e x)^n ]₁^e - ₁^e x n ( _e x)^ n-1 1 x , dx I_n = e (1)^n - 0 - n ₁^e ( _e x)^ n-1 , dx I_n = e - n I_ n-1 I_n + n I_ n-1 = e Given I_m + m I₂₀₂₆ = e Comparing the two equations, we get n = m and n-1 = 2026 . m - 1 = 2026 m = 2027