Calculus Facts_ Derivative of an Integral

8/3/2019 Calculus Facts_ Derivative of an Integral

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Calculus Facts

Derivative of an Integral (Fundamental Theorem of Calculus)

Using the fundamental theorem of calculus to find the derivative (with respect to x) ofan integral like

seems to cause students great difficulty. We'll try to clear up the confusion.

Here's the fundamental theorem of calculus:

Theorem If f is a function that is continuous on an open interval I, if a is any point in the

interval I, and if the function F is defined by

then the derivative of F(x) is F'(x) = f(x) for every x in the interval I.

(Sometimes this theorem is called the second fundamental theorem of calculus .)

Another way of stating the conclusion of the fundamental theorem of calculus is:

The conclusion of the fundamental theorem of calculus can be loosely expressed inwords as: "the derivative of an integral of a function is that original function", or"differentiation undoes the result of integration".

This description in words is certainly true without any further interpretation for indefinite integrals: if F(x) is an antiderivative of f(x), then:

Example 1: Let f(x) = x3 + cos(x). The (indefinite) integral of f(x) is

The derivative of this integral is



so we see that the derivative of the (indefinite) integral of this function f(x) is f(x).(Reminder: this is one example, which is not enough to prove the generalstatement that the derivative of an indefinite integral is the original function - it justshows that the statement works for this one example.)

Now the fundamental theorem of calculus is about definite integrals, and for a definite

integral we need to be careful to understand exactly what the theorem says and how itis used. Some of the confusion seems to come from the notation used in the statementof the theorem.

The first thing to notice about the fundamental theorem of calculus is that the variableof differentiation appears as the upper limit of integration in the integral:

Think about it for a moment. Unless the variable x appears in either (or both) of thelimits of integration, the result of the definite integral will not involve x, and so the

derivative of that definite integral will be zero. Here are two examples of derivatives ofsuch integrals.

Example 2: Let f(x) = ex -2. Compute the derivative of the integral of f(x) from x=0to x=3:

As expected, the definite integral with constant limits produces a number as ananswer, and so the derivative of the integral is zero.

Example 3: Let f(x) = 3x

2

. Compute the derivative of the integral of f(x) from x=0 tox=t:

Even though the upper limit is the variable t, as far as the differentiation withrespect to x is concerned, t behaves as a constant. So the derivative is again zero.

The result is completely different if we switch t and x in the integral (but still differentiatethe result of the integral with respect to x).

Example 4: Let f(t) = 3t2. Compute the derivative of the integral of f(t) from t=0 to

t=x:

This example is in the form of the conclusion of the fundamental theorem ofcalculus. We work it both ways. First, actually compute the definite integral and



take its derivative. Second, notice that the answer is exactly what the theorem saysit should be!

Note the important fact about function notation: f(x) is the same exact formula as f(t),except that x has replaced t everywhere.

The great beauty of the conclusion of the fundamental theorem of calculus is that it istrue even if we can't (easily, or at all) compute the integral in terms of functions we know! The theorem says that provided the problem matches the correct form exactly,we can just write down the answer . In Example 4 we went to the trouble (which was notdifficult in this case) of computing the integral and then the derivative, but we didn'tneed to. The theorem already told us to expect f(x) = 3x2 as the answer. It also tells us

the answer to the problem at the top of the page, without even trying to compute thenasty integral.

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Calculus Facts_ Derivative of an Integral