Why is a constant added in integration?

Answer 1

In order for your integration to be complete it has to represent the fact that it has infinite solutions, or else there's a possibility to have fallacies in the proofs you write. In other words, by neglecting the constant in your answer, it may be possible to extrapolate erroneous proofs from your incomplete answer, like 1 + 1 = 3, for example.

A little more practically, the constants of integration are great for bookkeeping when dealing with multiple integrals.

For example, here's the result from a simple triple integral without adding in the constants of integration:

∫∫∫ (xyz) dxdydz = ∫∫ (yzx2/2) dydz = ∫ (x2y2z/4) dz = x2y2z2/8

Whereas, with the constants added in you get this result:

∫∫∫ (xyz) dxdydz = ∫∫ (yzx2/2 + C) dydz = ∫ (x2y2z/4 + Cy + D) dz =

x2y2z2/8 + Cyz + Dz + E, where C, D, and E are the constants of integration.

This result has a term with both yz and z in it that we had initially missed, which could have had crucial applications to whatever this function is describing.