- #1
ddddd28
- 73
- 4
Is it possible to do an integral of f(x)*x without knowing f(x)?
ddddd28 said:Is it possible to do an integral of f(x)*x without knowing f(x)?
I am afraid that you've made a mistake.zinq said:Using integration by parts and letting u = x and dv = f(x)dx, we get
∫xf(x)dx = ∫udv = uv - ∫vdu = x∫f(x)dx - ∫f(x)dx
which is probably the most that can be said about the matter.
zinq said:Thank you for the correction! I should have written:
∫xf(w)dw = ∫xudv = uv]x - ∫xvdu = x∫xf(w)dw - ∫x(∫zf(w)dw)dz,
or maybe I should have just left it at
∫f(w)dw = uv - ∫vdu
and kept things simple.
zinq said:"What was wrong with x as the variable in the first place?"
To be technically correct, it's best to express an integral that ends up as being a function of (say) x in terms of integrating some variable that is not x (it doesn't matter which one). That's called a "dummy variable".
If the integral is indefinite and ends up to be a function of (say) x, then it needs an x, of course, and the proper place for the x is as a constant of integration. (Even though after the integral is taken, x need not be thought of as a constant.)
I hope that was sufficiently confusing (:-)>.
ddddd28 said:Is it possible to do an integral of f(x)*x without knowing f(x)?
zinq said:"Apart from a bit in the middle, that post is nonsense."
You are correct; I wrote "constant of integration" where I meant to say "limit of integration". I hope #7 no longer seems quite so nonsensical with that correction.
Yes, any function can be integrated with respect to x, as long as it is continuous. This means that there are no breaks or discontinuities in the function.
The purpose of integration is to find the area under a curve of a function. This can be useful in many applications, such as calculating volumes, distances, or work done.
To find the integral of f(x)*x, you can use the integration by parts method or the substitution method. These methods involve using algebraic manipulations and rules of integration to find the answer.
There is no specific formula for finding the integral of f(x)*x. The method used to find the integral may vary depending on the specific function and its complexity.
Yes, the integral of f(x)*x can be negative. This may occur if the function has negative values or if the area under the curve is below the x-axis. In this case, the integral would represent a negative value.