Do You Know, Why Do We Write “+c” After Integrating An Expression?

Everytime we integrate an expression, we write “+c” after it. But do you know, why we write this and where does it come from? Let, find the reason by this example:

Say you differentiate x²+5, the answer is 2x.
But if you differentiate x²+3 the answer is also 2x.
And if you differentiate x²-1000000 the answer is still 2x.

So if you integrate 2x the answer could be x²  or it could be x²+5 or x²+3 or x²-1000000.

So when we integrate 2x we give the answer as x²+c to cover all the possible answers.

This is because although when you differentiated c, it had no value, however if you integrate back, and exclude x, then it is not the same original equation that you had before (assuming c is not 0)

So for example let y = 2x +4, then dy/dx = 2, so if we integrate this, we expect to get back the original equation, i.e. y = 2x+4

However, if we ignore c, instead we get y = 2x, which is not the original equation, and thus this is an inconsistency.

So we always need to add a c, just in case there is a constant added onto the original equation.