I wish to calculate $\int_{0}^{\infty}dx\int_{0}^{xz}\lambda^{2}e^{-\lambda(x+y)}dy $
I compared my result, and the result with Wolfram when setting $\lambda=3$ and I get different results.
What I did:
$\int_{0}^{\infty}dx\int_{0}^{xz}\lambda^{2}e^{-\lambda(x+y)}dy $
$=\lambda^{2}\int_{0}^{\infty}\frac{e^{-\lambda(x+y)}}{-\lambda}|_{0}^{xz}\, dx$
$=\lambda^{2}\int_{0}^{\infty}\frac{e^{-\lambda(x+xz)}}{-\lambda}-\frac{e^{-\lambda x}}{-\lambda}\, dx$
$=-\lambda(\int_{0}^{\infty}e^{-\lambda(1+z)x}\, dx-\int_{0}^{\infty}e^{-\lambda x}\, dx)$
$=-\lambda(\frac{e^{-\lambda(1+z)x}}{-\lambda(1+z)}|_{0}^{\infty}-\frac{e^{-\lambda x}}{-\lambda}|_{0}^{\infty})$
$=-\lambda(\frac{1}{\lambda(1+z)}-(\frac{1}{-\lambda}))$
$=-\lambda(\frac{1}{\lambda(1+z)}+\frac{1}{\lambda})$
$=1-\frac{1}{1+z}$
I went over the calculation a couple of times and not only that I couldn't find my mistake, I also don't understand how I can end up with $1-e^{\text{something}}$ because the integrals are done from $0$ to $\infty$ and then I get $1$ or $0$ when I set the limits.
Can someone please help me understand where I am mistaken ?