e^ 2x 2( 2x 2x - 1) 2 ^2 2x , dx = A , e^ 2x 2x + c (Where c is the constant of integration.), then A^3 =

Step-by-step solution

Let the given integral be I . I = e^ 2x 2( 2x 2x - 1) 2 ^2 2x , dx I = e^ 2x ( 2x 2x ^2 2x - 1 ^2 2x ) , dx I = e^ 2x ( 2x - ^2 2x) , dx We know the standard integral form: e^ ax (a f(x) + f'(x)) , dx = e^ ax f(x) + c . Let a = 2 and f(x) = 1 2 2x . Then a f(x) = 2 ( 1 2 2x ) = 2x . Differentiating f(x) with respect to x , we get: f'(x) = 1 2 (- ^2 2x) 2 = - ^2 2x Substituting these into the integral, we get: I = e^ 2x (2 f(x) + f'(x)) , dx = e^ 2x f(x) + c I = 1 2 e^ 2x 2x + c Comparing this with the given express