# Tagged Questions

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### Integral $\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k))$$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
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### integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$$ It appears to be ...
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### Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
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### How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a})$. Could you tell me where did I do the mistake? if $x=z$ then ...
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### Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
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### Evaluate $\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$

Evaluate the following integral $$\int_0^{\frac{\pi}{2}}\frac{x^2}{1+\cos^2 x}dx$$ This function does not have an elementary anti-derivative. How can we solve this?
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### Integral $\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$ via contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some some ...
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### Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
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### find general solution to the Differential equation

Find the general solution to the differential equation $$\frac{dy}{dx}= 3x^2 y^2 - y^2$$ I get y=6xy^2 + 6x^2 y\frac{dy}{dx} - 2y\frac{dy}{dx} ...
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### Evaluating the integral $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx$
I don't know how to deal with this integral: $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx,$$ where z is a complex number.
Evaluate: $\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$. $\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$ $\int_C (z-z_0)^{n-1}dz$ for any integer $n$, ...