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
This yields:
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:
e^(ax)
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.
Personal Finance
To start solving the integral of xe^(-2x), you can use either integration by parts or differentiation with respect to a parameter. Let's first use the method of integration by parts.
1) Integration by parts:
Start by writing the integral as:
∫ xe^(-2x) dx
Next, use the formula for integration by parts:
∫ u dv = uv - ∫ v du
Let u = x and dv = e^(-2x) dx. Then, differentiate u to get du = dx and integrate dv to get v = (-1/2) e^(-2x).
Apply the integration by parts formula:
∫ xe^(-2x) dx = (-1/2) xe^(-2x) - ∫ (-1/2)e^(-2x) dx
Simplify the integral on the right-hand side:
∫ xe^(-2x) dx = (-1/2) xe^(-2x) + (1/4) e^(-2x) + C
So, the solution to the integral using integration by parts is (-1/2) xe^(-2x) + (1/4) e^(-2x) + C, where C is the constant of integration.
2) Differentiation with a parameter:
Consider the function e^(ax), where a is a parameter. The integral of this function is given by:
∫ e^(ax) dx = (1/a) e^(ax)
Now, differentiate both sides of the equation with respect to a. We want to find the derivative of the integral with respect to a:
d/dx [∫ e^(ax) dx] = d/dx [(1/a) e^(ax)]
Using the fundamental theorem of calculus, the left-hand side simplifies to:
e^(ax) = d/dx [(1/a) e^(ax)]
Now, substitute a = -2 to get the answer for the integral of xe^(-2x):
∫ xe^(-2x) dx = [-1/((-2)^2) + x/(-2)] e^(-2x)
Simplify the expression inside the square brackets and replace (-2)^2 with 4:
∫ xe^(-2x) dx = [-1/4 + x/(-2)] e^(-2x)
This gives the same result as the method of integration by parts: (-1/2) xe^(-2x) + (1/4) e^(-2x) + C.
In summary, you can solve the integral of xe^(-2x) either using integration by parts or differentiation with respect to a parameter. Both methods will lead to the same result: (-1/2) xe^(-2x) + (1/4) e^(-2x) + C.