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• 267k

### Integrate $\frac1{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$ over $[-1,1]^2$?

$$\int\frac{du}{\sqrt{u^2+v^2}\left(1+u^2+v^2\right)}=\frac{1}{\sqrt{v^2+1}}\tanh ^{-1}\left(\frac{u^2+v^2+1-u \sqrt{u^2+v^2}}{\sqrt{v^2+1}}\right)$$ Use the bounds and the logarithmic form of the ...
• 267k

• 22.3k

### Computing the integral $\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi.$

Restatement of the problem. Firstly, the logarithm can be split thanks to the given factorization, namely $(1 - 2x\cos\phi + x^2) = (1-xe^{i\phi})(1-xe^{-i\phi})$. Secondly, the factor $\phi^2$ can be ...
• 10.2k
Accepted

1 vote

### If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$?

If $h=0$ almost everywhere, then it has a null integral. (To prove this, try showing it for characteristic functions, and then use the definition of the integral).
• 319
1 vote

### Double integral of the form exp(-a(x-y)^2)

I'm gonna sketch out the solution, the questions are welcome in the comments: First, apply the change of variables in the following form:  \left\{ \begin{array}{} u&=& x-y \\ v&=& x+...
• 1,076

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