when you antidifferentiate 1/(2x+1)

[sozener1]

when you antidifferentiate 1/(2x+1)

I get 1/2ln(2x+1) do I also have to include the constant c??

so i beomes 1/2 ln (2x+1) + c ??

 

[HallsofIvy]

when you antidifferentiate 1/(2x+1)

I get 1/2ln(2x+1) do I also have to include the constant c??

so i beomes 1/2 ln (2x+1) + c ??

Yes, of course. To "anti-differentiate" (or, perhaps better, to "find the anti-derivative" of) function f, means to find all functions whose derivative is f. Since the derivative of a constant is 0, the derivative of (1/2)ln2+ 1)+c is 1/(2x+ 1)+ 0=1/(2x+ 1) for any c.

(You can use the mean value theorem to show that every anti derivative of 1/(2x+ 1) must be of that form- there are no others.)

 

[pka]

when you antidifferentiate 1/(2x+1)
I get 1/2ln(2x+1) do I also have to include the constant c??
so i beomes 1/2 ln (2x+1) + c ??

The rule-of-thumb is yes, include some constant.
It is a reminder that we only can find antiderivates 'up to a constant'.

 

[lookagain]

when you antidifferentiate 1/(2x+1)

I get 1/2ln(2x+1) do I also have to include the constant c??

so i beomes 1/2 ln (2x+1) + c ??$\displaystyle \ \ \ \ \$That is incorrect. You left off the required absolute value bars.

(1/2)ln|2x + 1| + C


$\displaystyle \int\frac{dx}{2x + 1} \ = \ \frac{1}{2}\ln|2x + 1| \ + \ C$

 

[Quaid]

when antidifferentiate 1/(2x+1)

I get 1/2ln(2x+1) do I also have to include the constant c??

Hi sozener:

The answer to your question is a function of why you antidifferentiated. :wink:

For example, if this is part of some exercise, then you need to scrutinize the exercise instructions, to determine whether they want the antiderivative (aka general antiderivative) versus an antiderivative.

On the other hand, if your antidifferentiation is one step in the solution process to some applied exercise, then it could be that any antiderivative is acceptable in your work.

By the way, be wary of a hidden trap; there are some general antiderivatives where C is actually a function of x (i.e., C takes on different values, in different intervals of the domain).

As an aside, I've seen no consistency on when and where instructors and authors abandon the +C, in subsequent notation or reference to the general antiderivative.

Same goes for using absolute-value symbols around natural-log inputs.

:cool: