A student asks: Why is there a +c when you integrate?

A student asks:

We’ve just started integration and I don’t understand why there’s always a $+c$ - I understand it’s a constant, I just don’t understand why it’s there!

Great question!

The simple answer is, because constants vanish when you differentiate, they have to appear when you integrate - it’s the opposite process.

If you think about straight lines, there are an infinite number of lines with a gradient of, say, 2: $y=2x$, $y=2x+4$, $y=2x-1$, $y=2x+113\pi$, and so on. All of those, when you differentiate them, give you $\frac{dy}{dx}=2$.

That means, when you integrate dy/dx with respect to dx, you get $y=2x$… plus something else, and you don’t know what it is unless you have a point on the line - so you just call it $c$ and work it out if you have the information.