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pivoxa15
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Homework Statement
Why are some integrals not able to be evaluated?
i.e. the integral of (1+1/x^4)^(1/2) is impossible to evaluate exactly from 0 to 1. Why??
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Hurkyl said:It's hard to answer your question without knowing just what you mean by "evaluated", "able", and "why".
HallsofIvy said:If you mean "find an expression for the anti-derivative in an elementary form", that is true for almost all integrable functions. the problem is that we simply don't know enough functions. By "elementary functions" we typically mean rational functions, radicals, exponentials and logarithms, and trig functions. That is just a tiny part of all possible functions, even all possible analytic functions.
In a deeper sense, the problem is that while we have "formula" for the derivative, there is no "formula" for the anti-derivative; it is simply defined as the "inverse" of the derivative. And "inverses" are typically very difficult. If we define [itex]y= x^5- 3x^3+ 4x^3- 5x^2+ x- 7[/itex], the direct problem, to "evaluate" the function (Given x, what is y?) is relatively simple. The "inverse" problem, to "solve the equation" (Givey y, what is x) is much harder
Given an analytic function to be integrated, we certainly can calculate its Taylor's series and integrate that term by term to get a power series for the anti-derivative. If I remember correctly, it should converge on the same radius of convergence as the original function.pivoxa15 said:So we will have an infinite series as an antiderivative for the function? If so then we can compute the series? It might diverge?
What precisely do you mean by "compute"?pivoxa15 said:evaluate by integrating from 0 to 1. which I have added in the OP.
able as you computing it exactly.
why as in why is it not able to be computed exactly.
The purpose of evaluating integrals is to calculate the exact area under a curve or the exact value of a function over a given interval. This is important in many fields of science, including physics, engineering, and statistics.
To evaluate integrals, you can use various techniques such as substitution, integration by parts, or trigonometric substitution. You can also use numerical methods such as the trapezoidal rule or Simpson's rule to approximate the value of an integral.
Knowing when to evaluate an integral is important because there are some cases where it is not possible to find an exact solution. In these cases, numerical methods can be used to approximate the integral, but this may result in some error. It is also important to understand the limitations of integration techniques to accurately evaluate integrals.
Evaluating integrals has many real-world applications, including calculating the area under a velocity vs. time graph to determine displacement, finding the volume of a three-dimensional object, and determining the probability of certain events in statistics.
Yes, there are some common mistakes to avoid when evaluating integrals. These include forgetting to apply the chain rule when using substitution, incorrectly applying integration by parts, and forgetting to account for limits when using numerical methods. It is also important to check your answer for correctness and to simplify as much as possible.