# Thread: Integration of two exponents

1. ## Integration of two exponents

Hi all!

I'm trying to solve a Laplace transform ($\displaystyle \mathfrak{L}(te^{kt})$, specifically). One process I tried ended up with the integral:

$\displaystyle \int_{0}^{\infty} e^{kt}e^{-st}dt$
$\displaystyle k \in \mathbb{R}$

I don't know how to solve this. At first glace, the product nature would suggest Integration by Parts but because both terms are exponents, this is recursive (although my integration by parts is sketchy - perhaps it is possible to use a substitution to clear that?).

Is this possible to integrate? For that matter, can a derivative be found? How?

Ta

2. Originally Posted by alexandicity
Hi all!

I'm trying to solve a Laplace transform ($\displaystyle \mathfrak{L}(te^{kt})$, specifically). One process I tried ended up with the integral:

$\displaystyle \int_{0}^{\infty} e^{kt}e^{-st}dt$
$\displaystyle k \in \mathbb{R}$

I don't know how to solve this. At first glace, the product nature would suggest Integration by Parts but because both terms are exponents, this is recursive (although my integration by parts is sketchy - perhaps it is possible to use a substitution to clear that?).

Is this possible to integrate? For that matter, can a derivative be found? How?

Ta
$\displaystyle e^{kt}e^{-st}=e^{-(s-k)t}$

Can you compute the integral now?

3. $\displaystyle \frac{1}{=-(s-k)}e^{-(s-k)t}+C$

I feel more than a little foolish for not seeing that - thanks Chris. Managed to solve the whole thing with that.

Although the next question is a little frustrating - "do it again, but differently!"