We know that $\int \frac{sin\frac{1}{x}}{x}$ cannot be expressed as in terms of elementary functions. Can the reason be given as like this :
Suppose it was integrable then we would have the derivative to be discontinuous at x=0 [limit doesnt exist there ] but we know that derivative of a function always follows intermediate value theorem hence there cannot be any jumps in the derivative , hence the function antiderivative doesnt exist ?