The following is Lemma 4.6. in Titchmarsh's book The Theory of the Riemann Zeta-Function:
My questions:
1- (Red-underlined) WolframAlpha gives $\int_0^{\infty} \dfrac{e^{iu}}{\sqrt{u}}$ same as in the text. For the remaining part, how $\int_{\frac12 {\delta^2} F''(c)}^{\infty} \dfrac{e^{iu}}{\sqrt{u}} = O \bigg( \dfrac{1}{\delta \sqrt{\lambda_2}} \bigg)$ is derived?
2- (Blue&Green-underlined) I could not understand the other two cases. For example, how the integral in the second line of the last paragraph is evaluated?