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List of integrals of exponential functions

One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. In this section, we examine exponential growth and decay in the context of some of these applications. Many systems exhibit exponential growth. Notice that in an exponential growth model, we have. That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. Notice that after only 2 hours minutesthe population is 10 times its original size! Note that we are using a continuous function to model what is inherently discrete behavior. At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values. When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling. Consider the population of bacteria described earlier. How many bacteria are present in the population after 4 hours? Interest that is not compounded is called simple interest. Compound interest is paid multiple times per year, depending on the compounding period. Mathematically speaking, at the end of the year, we have. Then we get.

List of integrals of exponential functions

Involving one direct function and elementary functions. Involving z alpha-1arguments b z and base a. Involving z alpha-1 and arguments a z. Involving z alpha-1arguments b z r and base a. Involving z alpha-1 and arguments a z r. Arguments involving inverse trigonometric functions. Arguments involving polynomials or algebraic functions and power factors. Arguments involving rational functions and power factors. With arguments c z r and base a. With arguments c z r p and base a. With arguments c z r p. Involving a d z h c z. Involving products of powers of the direct function. Involving product of power of the direct function and the direct function. Involving e c z e a z nu. Involving product of powers of two direct functions. Involving e c z mu e a z nu. Involving rational functions of the direct function. Involving algebraic functions of the direct function. Involving functions of the direct function and a power function. Involving powers of the direct function and a power function. Involving products of the direct function and a power function. Involving products of two direct functions and a power function. Involving z alpha-1 a d z h c z. Involving z alpha-1 a b z r h c z r. Involving products of powers of the direct function and a power function. Involving product of power of the direct function, the direct function and a power function.