how someone can know if it has an anti derivativeeys_physics said:I don't think this has an antiderivative in terms of elementary functions. However, for given limits you can compute it numerically.
rashida564 said:Homework Statement
What is the integral of e(x)2 over (x-1)2Homework Equations
integral of (x-1)-2 is -(x-1)-1
and derivative of e(x)2 is 2xe(x)2
The Attempt at a Solution
I tried to integrate it by part but I couldn't get a solution. I want to know how to start solving this question.
So sir any integral that involves e(x)2 can't be solvedRay Vickson said:I doubt that your integral can be performed in finite terms.
The simpler integral ##\int e^{x^2} \, dx## is known to be "non-elementary", which means that there is no possible finite formula for it that involves only only "ordinary functions". It does not matter if you allow a formula of 1,000,000 page length; as long as you do not use infinitely many pages you will not be able to do it. BTW: this is not a matter of nobody being smart enough to see how to do it; it is a rigorously proven theorem that it is impossible to do!
Of course, the integral exists, but the issue here is how to calculate it. There are numerous good approximations that allow us to get accurate numerical answers---even to hundreds of decimal places----so in practice we can get numbers easily enough.
rashida564 said:So sir any integral that involves e(x)2 can't be solved
This is not a "finite" formula, even though you can write it in a limited amount of space. When you expand it out you get a series that never ends.rashida564 said:can I write the answer as ∑(xn)/(n+1)n factorial from n=0 to infinity
The quotient rule in integration is a method used to find the antiderivative of a quotient function. It states that the integral of a quotient function f(x)/g(x) is equal to the integral of the numerator f(x) multiplied by the reciprocal of the denominator g(x).
The quotient rule in integration is used when the function being integrated is a quotient of two other functions. It is particularly useful when the numerator and denominator cannot be easily simplified or when the denominator contains variables.
No, the quotient rule in integration can only be used for functions that can be written in the form of f(x)/g(x), where f(x) and g(x) are functions of x. It cannot be applied to functions that are not in this form, such as trigonometric, exponential, or logarithmic functions.
The process for using the quotient rule in integration involves identifying the numerator and denominator functions, finding the antiderivative of the numerator and denominator separately, and then applying the quotient rule to obtain the final answer. It is important to simplify the resulting expression as much as possible.
Yes, there are a few special cases to keep in mind when using the quotient rule in integration. One is when the denominator is a constant, in which case the integral simplifies to the numerator multiplied by the reciprocal of the constant. Another is when the denominator is a linear function, in which case the integral can be found using a substitution method.