How would integrate $\frac 32e^{-x^2}$ without Gauss error function?

Question

I am going through my grade 12 textbook in the calculus section and I was asked to integrate this question: $$ \int_0^1 \frac 32 e^{-x^2}$$

I am in grade 12 and all the answers I find online require me to use the Gauss error function. I was wondering if there is another way of doing this. Thank you for your answers and sorry if this is a hopeless question.

Answer 1

Through power series, we will get:

$$e^{-x^2}= \displaystyle\sum_0^\infty \frac{(-1)^n\cdot x^{2n}}{n!}$$ Then, integrate term-by-term:

$$ \frac{3}{2}\cdot\displaystyle\int_0^1 \displaystyle\sum_0^\infty \frac{(-1)^n\cdot x^{2n}}{n!} dx $$

$$ \frac{3}{2}\cdot \displaystyle\sum_0^\infty \frac{(-1)^n}{n!} \displaystyle\int_0^1 x^{2n} dx $$

Which is the following convergent series

$$ \frac{3}{2}\cdot \displaystyle\sum_0^\infty \frac{(-1)^n}{n!(2n+1)} $$