Let x^{2} = t, Limit value x = [ 1 , 4 ] => t = [ 1 , 16 ]

And 2x dx = dt

=> 2dx / x = dt / x^{2} = dt / t.

∫ (2 e^{sinx^2} / x) dx

= ∫ (e^{sin t }) / t dt

= ∫ F ' ( t) dt // d/dx [f(x)] = e^{sinx }/ x , x > 0

= [F(t) ]_{1}^{16}

^{= }F(16 ) - F(1)

Ans - k= 16.