An identity to prove using calculus 1

[0kelvin]

Homework Statement

There is this function $$f(x) = e^{-A(x)} + e^{-A(x)} \int_0^x t e^{A(t)} dt$$

I have to prove that f satisfies $$f(x) + e^{-x^2} f(x) = x$$

Relevant Equations

$$A(x) = \int_0^x e^{-y^2} dy$$

I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.

 

[RPinPA]

I agree, I think one of those $f(x)$ terms in the thing you're supposed to prove is probably $f'(x)$

I would start by taking the derivative of $f(x)$ with respect to $x$. That means you're going to need the Fundamental Theorem of Calculus in a couple of places.

 

[0kelvin]

Ah right. $f'(x) + e^{-x^2}f(x) = x$.

Doing all the calculatiosn right, I'll end up with $xe^{A(x) - A(x)} = x$. Some chain and ´product rule here and there.

 

[WWGD]

Homework Statement: There is this function $$f(x) = e^{-A(x)} + e^{-A(x)} \int_0^x t e^{A(t)} dt$$

I have to prove that f satisfies $$f(x) + e^{-x^2} f(x) = x$$
Homework Equations: $$A(x) = \int_0^x e^{-y^2} dy$$

I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.

Sorry to nitpick, but , any conditions on A(x)? Continuity, differentiability, etc? Most likely even nicer, but just for the sake of completeness.

 

[0kelvin]

Teacher didn't specify. For some reason he thought it would be a good idea to give everyone 15 minutes to solve it last week. Solving it would give up to +1 point in the next exam.

 

[WWGD]

Teacher didn't specify. For some reason he thought it would be a good idea to give everyone 15 minutes to solve it last week. Solving it would give up to +1 point in the next exam.

But in order to differentiate, you would need to know something about A(t). I guess it is assumed that it is nice-enough to be differentiable.

 

[Delta2]

But in order to differentiate, you would need to know something about A(t). I guess it is assumed that it is nice-enough to be differentiable.

It is given that $A(x)=\int_0^x e^{-y^2}dy$ it is listed under homework equations tab in OP.

 

[WWGD]

It is given that $A(x)=\int_0^x e^{-y^2}dy$ it is listed under homework equations tab in OP.

My bad, did not pay attention.