1. Converging integrals

Show whether or not each of the following integrals converge:
i)

ii)

2. $
\int\limits_1^\infty {e^{ - \left( {3x^2 + 2} \right)} dx = e^{ - 2} } \int\limits_1^\infty {e^{ - 3x^2 } dx}

$

now it's easy to verify that:

$
e^{ - 3x^2 } \leqslant e^{ - 3x} \quad \forall 1 \leqslant x
$

thus:

$
\int\limits_1^\infty {e^{ - 3x^2 } } dx < \int\limits_1^\infty {e^{ - 3x} } dx\quad \forall 1 \leqslant x
$

now:

$
\int\limits_1^\infty {e^{ - 3x} } dx = \left. { - \frac{1}
{3}e^{ - 3x} } \right|_1^\infty = \frac{1}
{3}e^{ - 3} < \infty
$

thus according to the comparison test we conclude that the integral converges.

2.

note that the integrand is an even function, thus:

$
\int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} = 2\int\limits_0^\infty {e^{ - x^2 } dx}
$

now following the lines of the previous exercise we note that:

$
e^{ - x^2 } \leqslant e^{ - x} \quad \forall x \geqslant 0
$

....

3. Originally Posted by matty888

ii)
Since $e^x\ge x+1,\,\forall\,x\in\mathbb R$ we have $e^{x^{2}}\ge x^{2}+1\implies e^{-x^{2}}\le \frac{1}{x^{2}+1}.$

Hence $\int_{-\infty }^{+\infty }{e^{-x^{2}}\,dx}\le \int_{-\infty }^{+\infty }{\frac{1}{1+x^{2}}\,dx}=\pi .$ The given integral converges.

4. Originally Posted by Krizalid
Since $e^x\ge x+1,\,\forall\,x\in\mathbb R$ we have $e^{x^{2}}\ge x^{2}+1\implies e^{-x^{2}}\le \frac{1}{x^{2}+1}.$

Hence $\int_{-\infty }^{+\infty }{e^{-x^{2}}\,dx}\le \int_{-\infty }^{+\infty }{\frac{1}{1+x^{2}}\,dx}=\pi .$ The given integral converges.
nice