Indefinite Integrals

Definition (Indefinite Integrals) A function is called an antiderivative of a given function on an interval if for all in

Theorem (Indefinite Integrals) If is an antiderivative of the continuous function then any other antiderivative, of must have the form where is some constant.

    The notation where is an arbitrary constant means that is an antiderivative of It is called the indefinite integral of and satisfies the condition that for all in the domain of It is important to remember that represents a family of functions.

Example (Indefinite Integrals) Find the family of antiderivatives of the function and write an equation using the indefinite integral notation.

    Solution.  If then and so an antiderivative of sine is By the Antiderivative Theorem, the most general antiderivative is Therefore, where is an arbitrary constant.

Proposition (Indefinite Integrals) Let and be differentiable functions of such that Then



where is an antiderivative of This process is called integration by substitution.

Example (Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let then and by substitution, we have,
    










where is an arbitrary constant. Alternatively, we use and then and,








     

Example (Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let and then and so
    










where is an arbitrary constant.

Example(Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let then and then
    










where is an arbitrary constant.

Example (Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let then and
    












where is an arbitrary constant.

Example (Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let then and
    








where is an arbitrary constant.

Example (Indefinite Integrals) Use integration by substitution to evaluate

    Solution. Let then and
    










where is an arbitrary constant.  Alternatively, let then and
    

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