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magnifik
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Integrals of the exponential function
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In summary, the exponential function is a mathematical function used to model growth and decay in various fields. Integrals of this function are used in calculus to solve problems and can be found by replacing the function with its antiderivative. The exponential function and its integral are inverse functions and can be solved using integration by parts, although it is usually simpler to just use the fact that the integral is the original function.
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HallsofIvy
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What is the problem? You are told exactly what to do so do it! You are told that [itex]e^x\ge 1[/itex] for all [itex]x\ge 0[/itex] so
[tex]\int_0^x e^t dt\ge \int_0^x dt[/tex]
What does that give?
[tex]\int_0^x e^t dt\ge \int_0^x dt[/tex]
What does that give?
FAQ: Integrals of the exponential function
1. What is the exponential function?
The exponential function is a mathematical function with the form f(x) = ax, where a is a constant and x is the variable. It is commonly written as f(x) = e^x, where e is Euler's number (approximately 2.71828). The exponential function is used to model exponential growth and decay in various fields such as biology, physics, and finance.
2. What are integrals of the exponential function used for?
Integrals of the exponential function are used to solve various problems in calculus, such as finding the area under a curve or determining the change in a function over a given interval. They are also used in many applications, including physics, engineering, and economics.
3. How do you integrate the exponential function?
The integral of the exponential function is the antiderivative of the function itself. This means that to integrate the exponential function, you simply need to replace the function with its antiderivative, which is itself. Therefore, the integral of the exponential function is the original function, plus a constant of integration.
4. What is the relationship between the exponential function and its integral?
The exponential function and its integral are inverse functions of each other. This means that taking the integral and then the derivative of the exponential function will result in the original function. Similarly, taking the derivative and then the integral of the exponential function will also result in the original function.
5. Can integrals of the exponential function be solved using integration by parts?
Yes, integrals of the exponential function can be solved using integration by parts. This method involves breaking down the integral into two parts and using the product rule to solve it. However, in most cases, it is simpler to just use the fact that the integral of the exponential function is itself.