The ‘ symbol in c’ is not exactly the derivative in this case. Rather, it’s to denote a different constant.

Maybe using c1, c2 etc is better. The c’ probably attracts confusion amongst readers.

I see. Thank you

You'll have to get used to seeing ' in the questions , especially in the topic of Sets where there are things like A' and B', or labelling of certain points in geometry eg. A and A'.

Anyway the answer scheme has already defined c' as an arbitrary constant.


If one defines d as the constant in ∫2e^(-x) dx,

i.e ∫ 2e^(-x) dx = -2e^(-x) + d

then c' = d - c


So writing c' just means to group all the constants together simplify the expression

Alright but I have seen that instead of integrating exponential function, lnx seems to give no c' if you try to switch e with ln, is that a meaning to do this or am I wrong?

Different functions being integrated result in different Constants of integration. There will be an unknown constant no matter the integration.

So it's important to identify whether the constant has changed by substituting c with c'? But can I just put c instead? since constant is always different no matter what

The constant of integration in general will not be the same throughout, as is the case here. You can use almost every letter for the constant other than your chosen variables like x and y.

So in the first integration I can write +c, and after the second integration I can write an overall new constant k.

I mean I do find it confusing for me whenever I got to put c'

c1, c2 etc (subscript 1, 2 etc) is probably easier to see.

I now understand more about the meaning of arbitrary constant of integration. Thank so much. I rate 4.5/5

One last thing, all this changing the constant to a new one by writing different symbols like c',c1,c2 or even k except y and x can be regarded as a correct answer still when written in a real exam?

I recommend not using letters w, x, y and z as these are usually taken to be variables. The letter e should not be used also because it may be mistaken for exponential. Similarly, i should not be used as it may be mistaken for an “imaginary number” (square root of -1) which is learnt in the A Levels.

Having said that, we should stick to c as far as possible. The alternative ones are usually a, b, d and k.

Of course. Safe letters are usually c,d,a,b,k,l. If you run out of letters, use the ' sign

Your example :

∫ xe^(-x) = ∫ 2e^(-x) dx - xe^(-x) + e^(-x) - c

∫xe^(-x) dx = -2e^(-x) + d - xe^(-x) + e^(-x) - c

∫xe^(-x) dx = -e^(-x) + d - xe^(-x) - c

∫xe^(-x) dx = e^(-x)(-1 - x) + d - c

∫xe^(-x) dx = e^(-x)(-1 - x) + k

where k, d and c are constants and k = d - c

Alternative working for part ii)


∫xe^(-x) dx

= ∫ [(x - 2)e^(-x) + 2e^(-x) ] dx

= ∫ [ -(2 - x)e^(-x) + 2e^(-x) ] dx

= -(x - 1)e^(-x) + 2e^(-x) / (-1)

= -xe^(-x) + e^(-x) - 2e^(-x)

= -xe^(-x) - e^(-x)

= (-x - 1)e^(-x) + c, c is a constant




This way you avoid writing constant until the very end.

But it looks mathematically incorrect to not put the +c in the intermediary steps after the integration has been performed

I'm not sure what your teachers/school's style was, but leaving c to the very end was allowed for my time/institution for O levels.

Anyway, one can always write another c in the steps leading to the final answer

Sorry to trouble you guys, it seems the available slot I used to ask the question 2 weeks ago was apparently not answered even thought my doubts have been cleared. So can guys just help me and put the question as answered so I can sent out more questions in the future. Thanks

Erm for the tutors side we don't have such an option to mark your questions as answered. We can only comment or post answers. You might want to email the support team?

McDouble, possibly you can post on your own an answer stating “ok, done” text