Limit of improper integral $ \lim_{x\to \infty} \ \int_1^x x\,e^{t^2-x^2} \,dt$

Question

I need to calculate the limit of the following improper integral:

$$ \lim_{x\to \infty} \ \int_1^x x\,e^{t^2-x^2} \,dt$$

My solution is $\infty$, but when I enter it in WolframAlpha the solution is $\frac{1}{2}$, so I guess mine is wrong, but I can't figure out where. This is my current solution:

$$ First\ calculate\ the\ integral\ (leaving\ away\ the\ limit\ for\ now)\\[10pt] Let \ u = t^2-x^2 \\ \frac{du}{dt} = 2t \leftrightarrow dt = \frac{du}{2t} \\[20pt] \ Substitution:\\[5pt] \begin{align*} x \ \int_{1-x^2}^0 x\,e^{u} \frac{du}{2t} &= \frac{x}{2t} \ \int_{1-x^2}^0 e^{u} \,du \\ &= \frac{x}{2t} \ {\bigl (}{e^u{\bigl )}{\bigl \vert }\,}_{1-x^2}^{0} \\ &= \frac{x}{2t} \ {\bigl (}{e^0-e^{1-x^2}{\bigl )}\,} \\ &= \frac{x}{2t} - \frac{e^{1-x^2} \ x}{2t} \\ &= \frac{x - e^{1-x^2} \ x}{2t} \\ &= \frac{x \ (1-e^{1-x^2})}{2t} \end{align*}\\ \ \\[20pt] Now\ calculate\ the\ limit:\\[10pt] \lim_{x\to \infty} \ \frac{x \ (1-e^{1-x^2})}{2t} = \infty $$

What have I done wrong, that I don't get the solution $\frac{1}{2}$

Answer 1

It is $$ \int \limits_1^x x e^{t^2-x^2} dt = \frac{x}{e^{x^2}} \int \limits_1^x e^{t^2} dt = \frac{x}{e^{x^2}} \cdot (G(x) - G(1)) $$ for some differentiable function $G: [1, \infty) \rightarrow \mathbb{R}$ by the fundamental theorem of calculus. Furthermore, $G'(x) = e^{x^2}$, hence $G$ is strictly increasing and $\lim \limits_{x \to \infty} G(x) = + \infty$. Therefore, we are to calculate the limit $$ \lim \limits_{x \to \infty} \frac{x \cdot G(x)}{e^{x^2}} $$ This limit is of the form $\frac{\infty}{\infty}$, hence by L'Hospital's rule: $$ \lim \limits_{x \to \infty} \frac{x \cdot G(x)}{e^{x^2}} \overset{L'H}{=} \lim \limits_{x \to \infty} \frac{x \cdot G'(x) + G(x)}{2x \cdot e^{x^2}} = \frac{1}{2} + \lim \limits_{x \to \infty} \frac{G(x)}{2x \cdot e^{x^2}} $$ The second limit is again evaluated using L'Hospital: $$ \lim \limits_{x \to \infty} \frac{G(x)}{2x \cdot e^{x^2}} = \frac{1}{2} \lim \limits_{x \to \infty} \frac{G'(x)}{e^{x^2} + 2 x^2 \cdot e^{x^2}} = \frac{1}{2} \lim \limits_{x \to \infty} \frac{e^{x^2}}{e^{x^2} + 2 x^2 \cdot e^{x^2}} = \frac{1}{2} \lim \limits_{x \to \infty} \frac{1}{1 + 2 x^2} = 0 $$ All in all, we arrive at $$ \lim \limits_{x \to \infty} \int \limits_1^x x e^{t^2-x^2} dt = \lim \limits_{x \to \infty} \frac{x \cdot G(x)}{e^{x^2}} = \frac{1}{2} $$

Answer 2

When you do the substitution you cannot put the $1/t$ outside the integral. You should transform it into something with $u$ inside the integral.

Here is another way: The limit you want to calculate, is also the limit of the quotient $$ \frac{\int_1^x e^{t^2}\,dt}{e^{x^2}/x}. $$ Differentiating numerator and denominator separately, we get $$ \frac{\frac{d}{dx}\int_1^x e^{t^2}\,dt}{\frac{d}{dx}e^{x^2}/x}= \cdots= \frac{1}{2-1/x^2}. $$ The limit of the last expression, as $x\to+\infty$, is $1/2$. Thus, by the l'Hospital rule, this is also true for your original problem.