How do you find the constant of integration for #intf'(x)dx# if #f(2)=1#?

Answer 1
Normally, we want this integral function to be specified with a capital #f#, so that we can specify the antiderivative as #f(x)#.
However, using your variable naming, let's say that #F(x)# is the antiderivative of #f'(x)#, then by the Net Change Theorem, we have:
#f(x)=F(x)+C#

Therefore, the constant of integration is:

#C=f(x)-F(x)# #=f(2)-F(2)# #=1-F(2)#

This is a simple answer, however for many students, it is very difficult to this this abstractly. So, let's look at a concrete example:

#F(x)=x^3# to match your variables #F'(x)=f'(x)=3x^2# to match your variables #f(x)=int 3x^2 dx# #=x^3+C# #=F(x)+C#

Now, substitute the given values:

#f(2)=x^3+C=1# #2^3+C=1# #F(2)+C=1# #C=1-F(2)#

So, if an abstract problem makes it difficult for you to find a solution, start with a concrete one to help you find a pattern.

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Answer 2

To find the constant of integration for ∫f'(x) dx if f(2) = 1, you need to use the given information to determine the value of f(x) at x = 2. Once you know f(2), you can substitute it into the expression for ∫f'(x) dx and solve for the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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