The marginal price-demand function is the derivative of the price-demand function and it tells us how fast the price changes at a given level of production. In general, price decreases as quantity demanded increases. A price-demand function tells us the relationship between the quantity of a product demanded and the price of the product. Although the derivative represents a rate of change or a growth rate, the integral represents the total change or the total growth. The number e is often associated with compounded or accelerating growth, as we have seen in earlier sections about the derivative. As mentioned at the beginning of this section, exponential functions are used in many real-life applications. We can go directly to the formula for the antiderivative in the rule on integration formulas resulting in inverse trigonometric functions, and then evaluate the definite integral.
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We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let us first use a trigonometric identity to rewrite the integral. These two approaches are shown in Example. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution.Īlso, we have the option of replacing the original expression for u after we find the antiderivative, which means that we do not have to change the limits of integration. Substitution may be only one of the techniques needed to evaluate a definite integral. Use the process from Example to solve the problem. Although we will not formally prove this theorem, we justify it with some calculations here.
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If we change variables in the integrand, the limits of integration change as well. However, using substitution to evaluate a definite integral requires a change to the limits of integration. Substitution can be used with definite integrals, too.