How to calculate $\int_{\frac12 \eta}^{\infty} e^{-\frac{{\lambda}^2}{4 \pi}} {\lambda}^n d{\lambda}$?

Question

In the following text how the first blue-underlined O-term has changed to the second one?

I don't know how writing the integrand as $$e^{-\frac{{\lambda}^2}{8 \pi}} \times e^{-\frac{{\lambda}^2}{8 \pi}} {\lambda}^n$$ (up to a factor) helps the conclusion.

Source: Section 4.16. of Titchmarsh's book The Theory of the Riemann Zeta-Function.

Answer 1

Under the hypothesis mentionned, $$\int_{\eta/2}^\infty e^{-\lambda^2/(8\pi)}\times e^{-\lambda^2/(8\pi)}\left(\frac\lambda{\sqrt{2\pi}}\right)^nd\lambda\\\le\int_{\eta/2}^\infty e^{-\lambda^2/(8\pi)}\times e^{-\eta^2/(32\pi)}\left(\frac\eta{2\sqrt{2\pi}}\right)^nd\lambda\\\le e^{-\eta^2/(32\pi)}\left(\frac\eta{2\sqrt{2\pi}}\right)^n\int_0^\infty e^{-\lambda^2/(8\pi)}d\lambda\\ =O\left(e^{-\eta^2/(32\pi)}\left(\frac\eta{2\sqrt{2\pi}}\right)^n\right)\\ =O\left(e^{-t/16}\left(\frac{\sqrt t}2\right)^n\right). $$