I need to calculate the value of integral $\int_0^{\infty}(x^3-3x^2)\,e^{-4x}\,dx$. Counting it on $[0, 1]$ interval is simple using the trapezoidal or Simpson's rule.
Unfortunately I can't use those methods in this case because of the interval. I am not mathematical and I don't know all the useful methods to calculate integrals and other stuff. I am looking for the method which I can use to solve my problem.
Is there any similar method to Simpson's rule but for integrals in an infinite interval?
Do you need to use numerical methods? What about exact methods? First, use the following formula: $$\int_{\mathbb{R}^+}x^{n} e^{-ax}\,dx=\frac{\Gamma(n+1)}{a^{n+1}}$$ And applying it to your integral,
\begin{align} \int_0^{\infty} (x^2-3x^3)\,e^{-4x}\,dx&= \int_0^{\infty}x^2e^{-4x}\,dx - 3\int_0^{\infty} x^3 e^{-4x}\,dx \\ \\ &=\frac{2}{4^3}-3\left( \frac{6}{4^4}\right) \\ \\ &=\frac{1}{32}-\frac{18}{256} \\ \\ &=\frac{1}{32}-\frac{9}{128} \\ \\ &=-\frac{5}{128} \\ \\ &= -0.0390625 \end{align}