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General integration question

Just wanted to ask a question about integration. I am often unsure whether or not to add a constant c and try to find it when doing an integration question, eg if it asks to find the area under a curve, and when looking through mark schemes I have noticed that it varies, when are we meant to add a nd find a constant c ?

Thanks :smile:
When there aren't any limits.

You might want to read up on why you add the constant because then it will be obvious to know when to add it.
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dd1234
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Original post by electriic_ink
When there aren't any limits.

You might want to read up on why you add the constant because then it will be obvious to know when to add it.

Thanks, so is we only need the constant when doing antidifferentiation, not with integration ( limits)?
Original post by dd1234
Thanks, so is we only need the constant when doing antidifferentiation, not with integration ( limits)?


If you differentiate f(x) and f(x) + c with respect to x, you get the same thing. Hence, when you integrate without limits, it is necessary. It's acceptable to do it when you have limits but it just cancels out as demonstrated below

Suppose that

∫f(x)dx=F(x)+C\displaystyle \int f(x) dx = F(x) + C

Then

∫abf(x)dx=(F(b)+C)−(F(a)+C)=F(b)−F(a)\displaystyle \int_{a}^{b} f(x) dx = (F(b) + C) - (F(a) +C) = F(b)-F(a)

So the constant becomes redundant when you have limits.

Hope this helps :smile:
If you think of it like this and start with this equation.

y'(x)=2x

This is the gradient of the graph of y(x)=x^2, but can also be the gradient of y(x)=x^2+c

Notice that if you draw the graphs of y(x)=x^2+c for few different values of c, the gradient of each graph for one value of x will be the same. The constant is translating the graph up or down, so the gradient is not affected. Also, it's worth remembering that you get rid of any constants when differentiating, so that has to be taken into account because you're doing the opposite.

You can include the constant when you are using limits, but you will find that it will cancel after you do the subtraction after the squared brackets.
(edited 10 years ago)
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dd1234
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Thanks for the help! :smile:

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