I tried to integrate this:
intergral sign with an interval from -inifinity to + infinity
function: (x-r)^2 (1/r) (e^(-x/r))
with respect to dx.....
I know that Steve set up the u-substitution but I seem to never arrive at an answer... It seems like I keep integrating and integrating and integrating and never arrive at an answer!!!!!!!!
Steve gave you:
let
u = (x-r)^2
du = 2(x-r) dx
dv = (1/r) e^(-x/r) dx
v = e^(-x/r)
CHECK SIGN
∫ u dv = uv - ∫ v du
= (x-r)^2 e^(-x/r) - 2∫(x-r) e^(-x/r) dx
now the second part of it. He said to do it again.
u = x - r
du = dx
dv = e^-x/r
v = -r e^-x/r
-(x-r)e^-x/r - ∫-re^-x/r dx
now you can do ∫-re^-x/r dx = (1/r)e^-x/r
CHECK THE SIGNS !
then put it back together and do the limits
I got r^2 - 2r.. and it is wrong.... please help....
See whether you can get wolframalpha's result:
http://www.wolframalpha.com/input/?i=%E2%88%AB+%28%28x-r%29^2+%281%2Fr%29+%28e^%28-x%2Fr%29%29%29+dx
The limit would be undefined for x -> -∞
I suspect a typo somewhere. Seems like there should be e^(-x^2/r) or something.
It sounds like you are having trouble with the integration of the given function. Integrating this function requires a combination of techniques, including u-substitution and integration by parts.
To help you understand the process, let's go through the steps step by step:
Step 1: Apply u-substitution
Let's substitute u = -x/r. So, we have du = -dx/r. Now, we need to find the limits of integration when x is in the range from -∞ to +∞. When x = -∞, u would go from ∞ to 0, and when x = +∞, u would go from 0 to -∞.
Step 2: Simplify the expression
Now, substitute u back into the original function. We have:
(x - r)^2 (1/r) (e^(-x/r)) = ((-ru) - r)^2 (1/r) (e^u)
Simplify further: (r^2/u^2 + 2r/u +1) (e^u)
Step 3: Integrate
Let's focus on integrating each term separately:
The integral of r^2/u^2 with respect to u is r^2 * integral (1/u^2) du.
The integral of 2r/u with respect to u is 2r * integral (1/u) du.
The integral of 1 with respect to u is u.
The integral of e^u with respect to u is e^u.
After integrating each term, you need to substitute back u = -x/r and evaluate the limits of integration.
Step 4: Evaluate the limits of integration
Substitute the limits of integration back into the result from the previous step. Since we have the limits of integration as ∞ and -∞, it can be a bit more complicated to evaluate. One approach is to split the integral into two parts and evaluate them separately.
For example, let's split the integral into two parts:
∫ e^u du = ∫ e^u du from 0 to -∞ + ∫ e^u du from ∞ to 0
Now, plug in the limits of integration for each part and evaluate the integrals separately. Then combine the results, which might involve simplification or cancellation of terms.
Please note that due to the complexity of the function, it's not unexpected to have a lengthy integration process or a result that doesn't simplify nicely. Double-check your calculations, and if you're still uncertain, consider seeking additional guidance or using tools like computer software or online integrators to verify your answer.