expectation half-normal distribution or expectation Truncated Normal Distribution

Question

I want to calculate integrals $$ \begin{split} \int_0^\infty x \exp\{ ax-b x^2\} dx &= \int_0^\infty x\exp\{-b(x^2-\frac{a}{b}x)\}dx\\ &= \exp\left\{\frac{a^2}{4b}\right\}\frac{\sqrt{\pi}}{2} \int_0^\infty x \frac{\sqrt{2}}{\sqrt\pi(1/2)} \exp\left(\frac{-b}{2(1/2)}\left(x-\frac{a}{2b}\right)^2\right)dx \end{split} $$ Integral $$\int_{0}^{+\infty}x \frac{\sqrt{2}}{\sqrt\pi(\frac{1}{2})}\exp\frac{-b}{2(1\2)}(x-\frac{a}{2b})^2dx$$ is a expectation half-normal distribution or expectation Truncated Normal Distribution؟ and What is the final answer?