Double integral: $\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy$ Could someone help me with this question? I am stuck on it.

Compute the following double integral: $$\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy.$$

How to compute the integral when the inner integral is a form of error function? Thanks in advance for your help.
 A: The region of integration $y<x<1$ and $0<y<1$ is corresponding to $0<y<x$ and $0<x<1$, therefore
$$
\int_{y=0}^1\int_{x=y}^1\ e^{\large x^2}\ dx\ dy=\int_{x=0}^1\int_{y=0}^x\ e^{\large x^2}\ dy\ dx=\int_{x=0}^1\ e^{\large x^2}\ dx\ \int_{y=0}^x\ \ dy.
$$
The last integral is easy to be dealth with and the answer is $\color{blue}{\dfrac12(e-1)}$.
A: Another hard way (As the OP's wish):
Using Maclaurin series of exponential function, we will obtain
$$
e^{\large x^2}=\sum_{n=0}^\infty\frac{x^{\large 2n}}{n!}.
$$
Hence
$$
\begin{align}
\int_{x=y}^1\ e^{\large x^2}\ dx&=\int_{x=y}^1\ \sum_{n=0}^\infty\frac{x^{\large 2n}}{n!}\ dx\\
&=\sum_{n=0}^\infty\int_{x=y}^1\ \frac{x^{\large 2n}}{n!}\ dx\\
&=\left.\sum_{n=0}^\infty\frac{x^{\large 2n+1}}{(2n+1)\ n!}\right|_{x=y}^1\\
&=\sum_{n=0}^\infty\frac{1-y^{\large 2n+1}}{(2n+1)\ n!}
\end{align}
$$
and
$$
\begin{align}
\require{cancel}
\int_{y=0}^1\ \sum_{n=0}^\infty\frac{1-y^{\large 2n+1}}{(2n+1)\ n!}\ dy&=\sum_{n=0}^\infty\int_{y=0}^1\ \frac{1-y^{\large 2n+1}}{(2n+1)\ n!}\ dy\\
&=\sum_{n=0}^\infty\left.\left(\frac{y}{(2n+1)\ n!}-\frac{y^{\large 2n+2}}{(2n+2)(2n+1)\ n!}\right)\right|_{y=0}^1\\
&=\sum_{n=0}^\infty\left(\frac{1}{(2n+1)\ n!}-\frac{1}{(2n+2)(2n+1)\ n!}\right)\\
&=\sum_{n=0}^\infty\frac{(2n+2)-1}{(2n+2)(2n+1)\ n!}\\
&=\sum_{n=0}^\infty\frac{\cancel{2n+1}}{2(n+1)\cancel{(2n+1)}\ n!}\\
&=\frac12\sum_{n=0}^\infty\frac{1}{(n+1)!}\\
&=\large\color{blue}{\frac12(e-1)}.
\end{align}
$$
