# Tagged Questions

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### Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
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### Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
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### Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}}$$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
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### Convergence or divergence of the integral $\int_0^1 dx/\sin x$

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx$$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
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### Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Show that $$\int\limits^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$$ converge. I have utterly no clue on this integral. Please give me some hints. Thanks you.
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### Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
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### Showing the convergence of improper integral.

Hello I have to show that this improper integral is convergent: $$\int_{0}^{1} \frac{e^{\frac{-1}{x}}}{x^2} dx$$ , but I don't have any starting ideea. Any tips would be great, thank you.
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### Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx$$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
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### A simple convergent integral but not absolutely convergent.

Anybody knows a simple example for convergent function but not absolutely convergent? ( simple = easy ) Thanks for coments!!!
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### How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
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### How to calculate the improper integral $\int_0^\infty\left(\frac{1}{\sqrt{x^2+4}}-\frac{P}{x+2}\right)dx$

This is the first time I've seen a problem like this. I have no idea what to do. Detailed guidance would be of great help. For which values of P does the integral converge? ...
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### Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
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### Show that Improper integral converges

Show that the following integral is convergent. $$\int_{1}^{\infty}\sin\left(1 \over x^{2}\right)\cos\left(x^{2}\right)\,{\rm d}x$$ Not sure how I can solve ...
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### A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
### Discuss the convergence of $\int_0^1x^n \left[\log({1\over x})\right]^m \, dx$
Discuss the convergence of $$\int_0^1x^n\left[\log\left({1\over x}\right)\right]^m \, dx$$ Need some clues. I know that both $0$ and $1$ are points of discontinuities.