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We could potentially have a situation where we know that \begin{equation}  \int_0^\infty cos(x) \;dx - \int_0^\infty \frac{sin(x)}{x^2} \frac{sin(x)}{x}  \;dx = -\frac{\pi}{2} \end{equation}  That being the sum Which would give a specific value  of these two divergent integrals $0$ for the cos integral. However, it  is a constant well defined number. uncertain that one can trust this method.  Contradictory Some contradictory  examples to this it working  are \begin{equation}  \begin{array}{|c|c|}