Evaluating the Constant of Integration
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Key Questions

If
#F(x)# is an antiderivative of a function#f(x)# , that is,#F'(x)=f(x)# ,then
#G(x)=F(x)+C# , where#C# is any constant,is also an antiderivative of
#f(x)# since#G'(x)=[F(x)+C]'=F'(x)=f(x)# .Hence, there are a family of functions (only differ by a constant) that are antiderivatives of
#f(x)# . In order to include all antiderivatives of#f(x)# , the constant of integration#C# is used for indefinite integrals.#int f(x)dx=F(x)+C# The importance of
#C# is that it allows us to express the general form of antiderivatives.
I hope that this was helpful.

If
#G(x)# is an antiderivative of#f(x)# and#G(x_0)=y_0# , then#G(x)=int f(x)dx=F(x)+C# ,where
#F(x)# is any antiderivative of#f(x)# .Since
#G(x_0)=F(x_0)+C=y_0# ,we have
#C=y_0F(x_0)# .
I hope that this was helpful.

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)#
#=1F(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=1F(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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