how to integrate $e^{2t-\frac{3t^2}{2}}$

Question

I tried power series and got :

$$ \int\sum_{n=0}^\infty \frac{(4t-\frac{3}{2}t^2)}{n!}e^{2t-\frac{3t^2}{2}}dt$$

from there i got stuck.

Answer 1

Guessing you want $\int_{-\infty}^{\infty} \mathrm{e}^{4t - \frac{3}{2}t^2} \,\mathrm{d}t$.

• Complete the square in the exponent. This gives you a constant to move out of the integral.
• Make a linear change of variable to obtain $\int_{-\infty}^{\infty} \mathrm{e}^{-u^2} \,\mathrm{d}u$. This gives you a constant to move out of the integral.
• Having reduced the integral to a standard form, write down its value. $\int_{-\infty}^{\infty} \mathrm{e}^{-u^2} \,\mathrm{d}u = \sqrt{\pi}$.