there doesn't happen to be an integral evaluation rule for F(x)^G(x) does there?

A

Al-Layth

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is there a general rule for evaluating
[math]\int f(x)^{ g(x) } dx[/math]
 
pka said:
Suppose that each of [imath]f~\&~g[/imath] is a differentiable function.
If [imath]A(x)=\left[f(x)\right]^{g(x)}[/imath] then what is [imath]A^{\prime}(x)~?[/imath]
Click to expand...
I did some working and got:
[math]A'(x) = f(x)^{g(x)} \frac{f'(x)g(x)}{f(x)} +f(x)g(x)g'(x) \ln{f(x)}[/math]
but whats the point
isnt this the opposite of what i want lol
 
Al-Layth said:
I did some working and got:
[math]A'(x) = f(x)^{g(x)} \frac{f'(x)g(x)}{f(x)} +f(x)g(x)g'(x) \ln{f(x)}[/math]
but whats the point
isnt this the opposite of what i want lol
Click to expand...
Well of course that is not answering the question. But is shows what is needed to be present to get the anti derivative.
In other words [imath]\displaystyle\int {{x^3}\cos \left( {{x^4}} \right)dx}=[/imath][imath]\dfrac{\sin\left(x^4\right)}{4} [/imath] that is real easy being done by simply seeing the derivative. More complicated integrals require the construction of the antiderivative.
Consider a simple looking problem: [imath]\displaystyle\int {{{\left( {{x^2} - x} \right)}^{\sin (x)}}dx}[/imath]
See here Ther appears to be no closed form simple answer. But there are Definite integrals solutions.
[imath][/imath]
 
Al-Layth said:
is there a general rule for evaluating
[math]\int f(x)^{ g(x) } dx[/math]
Click to expand...
You've asked a number of questions of this same type, to which the answer is the same: There is no such general rule. You should have picked up the pattern by now.

Integration is harder than differentiation, in a way similar to the fact that division is harder than multiplication. There are simple rules for the "forward" operations, but not for the "reverse". To divide, you need to recognize exact multiples in the dividend (often by guessing); to find an antiderivative, you need to recognize derivatives of familiar functions within the integrand.

We teach a few rules for differentiation, with which you can differentiate just about anything. We teach only a couple tricks for integration (substitution and parts), each of which requires recognizing a pattern, and does not always produce anything useful. If there were rules for cases like those you have asked about. they would be in the books! The fact that they are not implies that they don't exist, because we would love to have them.

(But of course, we will not generally tell you directly that there can't be such a rule (see #2), because we don't like to say things we can't prove. But it is a virtual certainty.)
 
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