224 views

### Integral $\int_0^{\infty} \frac{e^{-x^2}}{a+b\cos{x}}dx$

Hello there I am trying to solve for $a > b$: $$I=\int_0^{\infty} \frac{e^{-x^2}}{a+b\cos{x}}dx$$ My thought was to expand into fourier series $$g(t)=\frac{1}{a+b\cos t}$$ Since g(t) has the ...
62 views

### Suitable contour for an integral ($\Gamma(1/2)$)

Consider the following integral $$\int_0^\infty \frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi}$$ This can be evaluated using contour integration methods. A similar question was asked before (unfortunately I ...
121 views

138 views

### Integrating this complicated integral for statistics [duplicate]

I want to show that : $$\int_{-\infty}^{\infty} e^\frac{-u^2}{2} du = \sqrt{2\pi}$$ Is there an elementary way using the tools of Calculus II to do this type of integration? I have not studied ...
1k views

### How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$? [duplicate]

How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$ using polar coordinates?
### Evaluating $\int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx$
Q: Suppose $\alpha>0$ and $|\beta|<\pi/2$, show that \begin{align*} \textbf{(1)} \; \int_0^{\infty}e^{-\alpha x^2 \cos \beta} \cos(\alpha x^2 \sin \beta) dx &= \frac 1 2 \sqrt{\pi/\alpha}\...