How to solve an integration problem involving exponential functions?

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How do i solve
\int_{0}^{\infty} \lambda^2 r e^{-\lambda r} dr

if i were to integrate it i get
\left[-e^{-lambda r} (1 + r \lambda)\right]_{0}^{\infty}

what is \lim_{r \rightarrow \infty} \frac{r}{e^r}

is it zero?? by virtue of e increasing faster than r ??
 
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The exponential increases faster than any polynomial.

Daniel.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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