When you integrate a function why do you add a constant?

That is a good question. Let me try to help you figure this out by working through a few simple examples. We know that differentiation is like a reverse process of right? So let us differentiate a few functions.

What is the derivative of f(x) = x^2 , f'(x)=2x ,right?

What is the derivative of f(x) = x^2 + 5, also f'(x)=2x, right?

What is the derivative of f(x) = x^2 + 10, also f'(x)=2x, right?

As you see the derivative is the same for all the function above. This is because differentiation gets rid of any constant given, meaning any value with no power of x in front of it disappears. Therefore, we add a constant when we integrate as we do not know what the exact function is, we just know what the coefficients of x are. Here is are diagrams to help understand this.