If x x ( x^ x^2 e^ - x^2 2 + x^ -x^2 e^ x^2 2 ) dx = 1 k ( x^ x^2 e^ x^2 - x^ x^2 e^ x^2 ) + C , where k > 0 and C is the constant of integration, then the value of k | |⁻¹ is

Step-by-step solution

Let I = x x ( x^ x^2 e^ - x^2 2 + x^ -x^2 e^ x^2 2 ) dx Let u = x^ x^2 e^ - x^2 2 . Taking the natural logarithm on both sides: u = x^2 x - x^2 2 Differentiating with respect to x : 1 u du dx = 2x x + x^2 ( 1 x ) - x = 2x x + x - x = 2x x x x u = 1 2 du dx Similarly, let v = x^ -x^2 e^ x^2 2 . Taking the natural logarithm: v = -x^2 x + x^2 2 Differentiating with respect to x : 1 v dv dx = -2x x - x^2 ( 1 x ) + x = -2x x - x + x = -2x x x x v = - 1 2 dv dx Substitute these into the integral: I = ( 1 2 du dx - 1 2 dv