# Taking the integral of xe^ax^2

1. Dec 21, 2011

### ilikesoldat

1. The problem statement, all variables and given/known data
dy/dx=x*e^(ax^2)

solve the differential equation

2. Relevant equations
integral of e^x=e^x

3. The attempt at a solution
im not really sure how to do it when there are two variables in the exponent? i tried several things like u=x^2 1/2du=xdx then 1/2*int[e^a*u] results, but i can't take the integral without a different substitution since int[e^a*u] is not = to e^a*u.

the answer is 1/2a * e^ax^2 + C.. any methods to integrate this? several methods would be best

2. Dec 21, 2011

### Mentallic

Think about how you compute $$\frac{d}{dx}e^{f(x)}$$

3. Dec 21, 2011

### Highway

a is a constant, keep that in mind . . .

4. Dec 21, 2011

### ilikesoldat

I solved it with a as a constant and got the right answer, but . . .

How do i know a is a constant? I can't just assume that can I? What is the proof for a being a constant

Last edited: Dec 21, 2011
5. Dec 21, 2011

### Mentallic

Because a is assumed to be a constant in these situations, just as how $\pi$ nearly always represents the irrational number, but in some other cases, it can be mean something completely different, such as representing a product.

And if it wasn't a constant, then you can't solve the problem. So which do you think is more likely? :tongue:

6. Dec 22, 2011

### HallsofIvy

Staff Emeritus
Was the question just "integrate $xe^{ax}$" or was it to find $\int xe^{ax}dx$?

7. Dec 22, 2011

### ilikesoldat

That makes sense but it still bugs me to just assume it is a constant :(

And to the other person, the question was

dy/dx=x*e^(ax^2)

solve the differential equation, so yes it would be integral of x*e^ax^2 dx (except you forgot the squared on top of the x in e's exponent)

thanks for everybody's help

8. Dec 22, 2011

### epenguin

The situation is no different than if you were told to solve dy/dx = a .

9. Dec 22, 2011

### Mentallic

What's the difference?

10. Dec 22, 2011

### HallsofIvy

Staff Emeritus
The second, $\int xe^{ax^2} dx$, makes it explicit that the 'variable of integration' is x ("dx") while the first, "integrate $xe^{ax^2}$", is ambiguous. Of course, it is a standard convention that such things as "x", "y", "z" are used as variables while such things as "a", "b", "c", etc. are used to denote constants.

If it were $\int xe^{ax^2}da$, it would be $(1/x)\int e^{ax^2}+ C$.

11. Dec 22, 2011

### Mentallic

Yes, but without those standard conventions then we could just as well argue that a=f(x) in the integrand, just as the OP has done.