How to evalutate this exponential integral

Question

Is there an easy way to compute $$\int_{-\infty}^\infty\exp(-x^2+2x)\mathrm{d}x$$ without using a computer package?

Answer 1

This is a Gaussian integral. In general you can use the formula $\int_{-\infty}^{\infty} \exp(-x^2+bx+c)\mathrm{d}x=\sqrt{\pi}~\exp(b^2/4+c)$. This formula, as suggested by Thomas, can be derived by completing the square in the exponent, and using $\int_{-\infty}^{\infty} \exp(-x^2)\mathrm{d}x=\sqrt{\pi}$.