# How do you show whether the improper integral int (x^2)(e^(-x^3)) dx converges or diverges from negative infinity to infinity?

Oct 8, 2015

See the explanation section, below.

#### Explanation:

We need (among other things):

${\lim}_{u \rightarrow - \infty} {e}^{u} = 0$ and ${\lim}_{u \rightarrow \infty} {e}^{u} = \infty$.

Let's note that $\int {x}^{2} {e}^{- {x}^{3}} \mathrm{dx} = - \frac{1}{3} {e}^{- {x}^{3}} + C$.
(By substitution with $u = {x}^{3}$.)

We also note that to (attempt to) evaluate an integral that is improper at both limits of integration, we need to breack the interval into two pieces using ${\int}_{a}^{b} f \left(x\right) \mathrm{dx} = {\int}_{a}^{c} f \left(x\right) \mathrm{dx} + {\int}_{c}^{b} f \left(x\right) \mathrm{dx}$.

Let's use $c = 0$, (Because that makes the exponential easy to evaluate.)

${\int}_{-} {\infty}^{\infty} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx} = {\int}_{-} {\infty}^{0} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx} + {\int}_{0}^{\infty} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx}$ $\text{ }$ (If both integrals exist.)

${\int}_{-} {\infty}^{0} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx} = {\lim}_{a \rightarrow - \infty} {\int}_{a}^{0} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx}$

 = lim_(ararr-oo) ((-1/3e^(-x^3))]_a^0)

$= {\lim}_{a \rightarrow - \infty} \left(- \frac{1}{3} + \frac{1}{3} {e}^{- {a}^{3}}\right)$

As $a \rightarrow - \infty$, the exponent $- {a}^{3} \rightarrow \infty$ so the integral diverges.

We conclude that the original integral, ${\int}_{-} {\infty}^{\infty} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx}$ diverges.

Note
By the way, the other integral, ${\int}_{0}^{\infty} \left({x}^{2}\right) \left({e}^{- {x}^{3}}\right) \mathrm{dx} = \frac{1}{3}$

Also, we might have noted before integrating, that
as $x \rightarrow - \infty$,
the integrand ${x}^{2} {e}^{- {x}^{3}} \rightarrow \infty$.
So. there was no chance of the integral left of zero being finite.

Here is the graph of the function:

graph{x^2 e^(-x^3) [-12.87, 32.78, -4.08, 18.77]}