Posted by walex on Tuesday, May 15, 2007 at 7:55pm.
how do you start this problem:
integral of xe^(-2x)
There are two ways:
1) Integration by parts.
2) Differentiation w.r.t. a suitably chosen parameter.
Lets do 1) first. This is the "standard method", but it is often more tedious than 2)
You first write the integral as:
Inegral xe^(-2x) dx =
Integral -1/2 x d(e^(-2x))
Here we have used that:
d(e^(-2x)) = -2 e^(-2x)
The next is is to make use of the fact that:
d(f g) = f dg + g df --->
f dg = d(fg) - g df
Integral -1/2 x d(e^(-2x)) =
Integral d[-1/2 x e^(-2x)] -
Integral -1/2 e^(-2x) dx =
-1/2 x e^(-2x) - 1/4 e^(-2x) + C
Method 2) is much simpler. Consider the function:
It's integral is:
Integral e^(ax)dx = 1/a e^(ax)
Le's differentiate both sides w.r.t. a:
Integral x e^(ax)dx =
[ -1/a^2 + x/a] e^(ax)
And insert a = -2 to obtain the answer.
Answer this Question
- integration by parts - s- integral s ln (2x+1)dx ? = ln(2x+1)x - s x d( ln (2x+1...
- Calculus - Hello, I have some calculus homework that I can't seem to get started...
- calc - evaluate the integral: y lny dy i know it's integration by parts but i ...
- Math/Calculus - How would I integrate the following by parts: Integral of: (x^2...
- Calc 2 - a. Integral (x^2)/(sqrt(1+(x^2))) Would I separate these two into 2 ...
- calc asap! - can you help me get started on this integral by parts? 4 S sqrt(t) ...
- Calculus - Use integration by parts to find the integral. Round the answer to ...
- Calculus II - Evaluate the integral using method of integration by parts: (...
- Stats: Joint Density Function - Let the joint density function of X and Ybe ...
- mathematics - what does 'dx' signify in the integral and differentiation ...