I'm trying to find P(X<0.5) for X~N(0,1) using integration without technology or polynomials to help with the integrals. Seems like I learned this method in grad school, but can't remember the last steps and haven't found anything online about it yet. I have: \begin{align} P(X<0.5) & = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{0.5} e^{-x^2/2}dx \\ & = 0.5 + \frac{1}{\sqrt{2\pi}} \sqrt{\int_{0}^{0.5}e^{-x^2/2}dx \int_{0}^{0.5}e^{-y^2/2}dy} \\ & = 0.5 + \frac{1}{\sqrt{2\pi}} \sqrt{\int_{0}^{0.5}\int_{0}^{0.5}e^{\frac{-(x^2+y^2)}{2}}dxdy} \\ & = 0.5 + \frac{1}{\sqrt{2\pi}} \sqrt{\int_{0}^{\pi/4}\int_{0}^{0.5\sec(\theta)}e^{\frac{-r^2}{2}}rdrd\theta + \int_{\pi/4}^{\pi/2}\int_{0}^{0.5\csc(\theta)}e^{\frac{-r^2}{2}}rdrd\theta} \\ & = 0.5 + \frac{1}{\sqrt{2\pi}} \sqrt{\int_{0}^{\pi/4}1-e^{\frac{-(0.5\sec(\theta))^2}{2}}d\theta + \int_{\pi/4}^{\pi/2}1-e^{\frac{-(0.5\csc(\theta))^2}{2}}d\theta} \end{align}
Dunno where to go from there. Help please.