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.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## Cav

Surely you aren’t implying touch treat a differential equation of a,straight line as having one constant? That’s a little close to the “y=mx+c” wire isn’t it? I thought you rolled with two…

## Colin

Me? Goodness, no, that’s the Mathematical Ninja. He’ll be very disappointed you mixed us up! I’d, uh, tread carefully if I were you 😉