Often times it so happens that the point of interest is not the numerical value of the integral but its behavior at different points in its domain. If I am able to figure out the expression it becomes easier.
To give an example consider the moment generating transform, Laplace transform. Their symbolic expression can be very informative.
Consider the Mercator projection. It was designed without any idea of the closed form of the required integral. It was mostly done by estimate and gut feel. Now that we know the actual form (an entirely serendipitous discovery) we feel more confident that we understand the transform. This part is considerably psychological but not entirely.
Note that when drawing a map in Mercator projection we have to fall back to numerical estimation. But it helps that parts of the transforms are built from functions tha have names, that means we have seen the same functions elsewhere, it instills a sense of familiarity and understanding.
There are way to many functions to name, so the ones we have given names to are a bit special.
To give an example consider the moment generating transform, Laplace transform. Their symbolic expression can be very informative.
Consider the Mercator projection. It was designed without any idea of the closed form of the required integral. It was mostly done by estimate and gut feel. Now that we know the actual form (an entirely serendipitous discovery) we feel more confident that we understand the transform. This part is considerably psychological but not entirely.
Note that when drawing a map in Mercator projection we have to fall back to numerical estimation. But it helps that parts of the transforms are built from functions tha have names, that means we have seen the same functions elsewhere, it instills a sense of familiarity and understanding.
There are way to many functions to name, so the ones we have given names to are a bit special.