# Tagged Questions

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### $\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty$ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
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### a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
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### Integral $\int\sqrt{\sin2x}\operatorname d\!x$

I tried all substitutions but failed. I need assistance to evaluate that indefinite integral. $\int\sqrt{\sin2x}\operatorname d\!x$
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### Find Indefinite of root function

I don't know how to find this strange integral $\int{\sqrt{\dfrac{x-4}{x+2}}\dfrac{dx}{x+2}}$ Please help me solve this problem
### If $f:[0,1]\rightarrow\mathbb{R}$ is a function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$?
If $f:[0,1]\rightarrow\mathbb{R}$ is a Riemann-Integrable function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$? I'm thinking about it but without progress. Would someone ...