Evaluating the Constant of Integration

Key Questions

  • How do I evaluate constants of integration?

    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_0-F(x_0).

    I hope that this was helpful.

    Wataru · · Oct 17 2014
  • How do you find the constant of integration for intf'(x)dx if f(2)=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.

    Amory W. · · Aug 19 2014
  • What is the constant of integration and why is it so important?

    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.

    Wataru · · Oct 17 2014

Questions

Techniques of Integration