Chapter 5 | Integration
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5.6 | Integrals Involving Exponential and Logarithmic Functions Learning Objectives
5.6.1 Integrate functions involving exponential functions. 5.6.2 Integrate functions involving logarithmic functions.
Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. In this section, we explore integration involving exponential and logarithmic functions. Integrals of Exponential Functions The exponential function is perhaps the most efficient function in terms of the operations of calculus. The exponential function, y = e x , is its own derivative and its own integral.
Rule: Integrals of Exponential Functions Exponential functions can be integrated using the following formulas.
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∫ e x dx = e x + C ∫ a x dx = a x ln a + C
Example 5.37 Finding an Antiderivative of an Exponential Function
Find the antiderivative of the exponential function e − x .
Solution Use substitution, setting u =− x , and then du =−1 dx . Multiply the du equation by −1, so you now have − du = dx . Then, ∫ e − x dx =− ∫ e u du =− e u + C =− e − x + C .
Find the antiderivative of the function using substitution: x 2 e −2 x 3 .
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A common mistake when dealing with exponential expressions is treating the exponent on e the same way we treat exponents in polynomial expressions. We cannot use the power rule for the exponent on e . This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. In these cases, we should always double-check to make sure we’re using the right rules for the functions we’re integrating.
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