When EVALUATING an indefinite integral, what must you not forget?
When evaluating an indefinite integral, there are a few important things that you must not forget:
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When evaluating an indefinite integral, there are a few important things that you must not forget:
1. Constant of integration: An indefinite integral represents a family of functions, so it is important to include the constant of integration when evaluating the integral. This constant arises because when we differentiate a function, the derivative loses any constant terms. Therefore, any antiderivative (indefinite integral) of a function should include this constant to indicate the family of functions that it belongs to.
For example, if you find the antiderivative of the function f(x) = 2x, you would write F(x) = x^2 + C, where F(x) is the indefinite integral of f(x) and C is the constant of integration. The inclusion of + C shows that there are countless antiderivatives of 2x, all of which differ by a constant value.
2. Limits of integration: When you are evaluating an indefinite integral, you often need to include limits of integration. The limits specify the interval over which you want to find the antiderivative. The general notation for evaluating an indefinite integral with limits of integration is:
∫ [lower limit, upper limit] f(x) dx
For example, if you want to find the antiderivative of f(x) = 2x^3 between x = 1 and x = 3, you would write:
∫ [1, 3] 2x^3 dx
In this case, you would first find the antiderivative of 2x^3 and then evaluate it at the upper limit (x = 3) and subtract its value at the lower limit (x = 1).
3. Algebraic manipulation: Before evaluating an indefinite integral, you may need to perform some algebraic manipulations to simplify the integrand. This can involve using algebraic identities, factoring, or expanding expressions. Simplifying the integrand can make integration easier and lead to a more manageable antiderivative.
It is important to remember all these aspects when evaluating an indefinite integral to ensure that your final result is correct and complete.
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