Value of $ \lim_{n \to \infty} \int \limits_{0}^{1}nx^n e^{ x^2} ?$ How to find the value of $$ \lim_{n \to \infty}\int \limits_{0}^{1} nx^n e^{ x^2} ?$$
From wolfram the limit approaches to $e$ for larger values of $n$. I substituted $x^2 $ with $u$ and obtained 
$$ \frac{ n} {2} \int \limits_{0}^{1} u^{\frac{n-1}{2}} e^{u} du $$ 
The value of this integral can be obtained from  here. But still I'm unable to get it. Is there any better approach for this question?

You have the right idea about changing the variable, just a different change: $u=x^{n+1}$
$$
\begin{align}
\lim_{n\to\infty}\int_0^1nx^ne^{x^2}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^1\frac{n}{n+1}e^{u^{\frac2{n+1}}}\,\mathrm{d}u\\
&=\int_0^11\cdot e^1\,\mathrm{d}u\\[6pt]
&=e
\end{align}
$$
Note that $\frac{n}{n+1}e^{u^{\frac2{n+1}}}$ increases monotonically to $e$ for all $u\in(0,1]$, and uniformly on compact subsets, so we can use monotone convergence, dominated convergence, or uniform convergence (on each compact subset) to justify the exchange of limit and integral.
