![]() evaluate the antiderivative at the lower limit.evaluate the antiderivative at the upper limit.find an antiderivative of the integrand.To determine the constant C observe thatĪt last we have an efficient method for evaluating definite integrals: Since F is also an antiderivative of f, it must be that F and G differ by (at most) a constant:įor some value of C. Define the function G on to beīy the First Fundamental Theorem of Calculus, G is an antiderivative of f. This topic will be revisited again in the Evaluating Definite Integrals lesson. The First Fundamental Theorem of Calculus tells us that an antiderivative for exists (for > 0) and can be defined in terms of a definite integral. Note that is the one power function for which we do not know an antiderivative. Now, differentiate using the First Fundamental Theorem of Calculus and the Chain Rule to obtain: ![]() To put this problem in a form where the general result in Example 2 can be used, rewrite the definite integral as The computation of this derivative is similar to the previous example except that there are functions on both limits of integration. For 0, > 0 and the integral is positive. Ifįirst, note that and are both positive when > 0. To conclude this example, there is a general pattern for differentiating definite integrals when the upper limit is a function. Also, the in the denominator of the derivative means that the oscillations decrease as increases. The function has local extrema at every zero of the cosine function, i.e., for = 0, 1. Notice that this result is consistent with the plot created at the beginning of this example. The derivative of G is obtained from the First Fundamental Theorem of Calculus: To compute the derivative, it is convenient to rewrite the function as Before starting to compute the derivative, let's see a plot of the function. This function is a little more complicated. Find a formula for the derivative,, that is valid for all >= 0. The equation that selects the correct antiderivative isĮq1 := eval( value( FF1 ) + C, x=1 ) = 0:ĭefined for all >= 0. This condition will be satisfied by exactly one antiderivative. To determine the appropriate value for, note that This means that the function F(x) must be one of antiderivatives of : (These values exist because f is continuous on this interval.) Then,īy the First Fundamental Theorem of Calculus, it is known that F is a differentiable function andĪt the same time, we know that the family of antiderivatives of Let and denote the largest and smallest values of f on the interval.
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