Write a value

We know that $\mathrm{e}^{\mathrm{a}+\mathrm{b}}=\mathrm{e}^{\mathrm{a}} \mathrm{e}^{\mathrm{b}}$

$y=\int e^{2 x^{2}} e^{\ln x} d x$

$y=\int e^{2 x^{2}} x d x$

Let, $x^{2}=t$

Differentiating both sides with respect to $x$

$\frac{d t}{d x}=2 x$

$\Rightarrow \frac{1}{2} d t=x d x$

$y=\int \frac{1}{2} e^{2 t} d t$

Use formula $\int e^{a+b t}=\frac{e^{a+b t}}{b}$

$y=\frac{1}{2} \frac{e^{2 t}}{2}+c$

Again, put $\mathrm{t}=\mathrm{x}^{2}$

$y=\frac{e^{2 x^{2}}}{4}+c$