If e^ ⁻¹ x ( 2x + (1+x^2) 1+x^2 ) dx = g(x) e^ ⁻¹ x + C , where C is the constant of integration, and g(0) = 0 , then g(1) is equal to :

Step-by-step solution

Let I = e^ ⁻¹ x ( 2x + (1+x^2) 1+x^2 ) dx . Substitute t = ⁻¹ x x = t and dx = ^2 t dt = (1+x^2) dt . The integral transforms to: I = e^t ( 2 t + (1+ ^2 t) 1+ ^2 t ) (1+ ^2 t) dt I = e^t (2 t + ( ^2 t)) dt I = e^t (2 t + 2 ( t)) dt Let f(t) = 2 ( t) . Differentiating with respect to t gives: f'(t) = 2 ( 1 t ) ( t t) = 2 t . Using the standard integral form e^t (f(t) + f'(t)) dt = e^t f(t) + C , we obtain: I = e^t (2 ( t)) + C Substituting back t = ⁻¹ x , we note that 2 ( t) = ( ^2 t) = (1+ ^2 t) = (1+x^2) . Thus, I