Tagged Questions

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How do I compute this improper integral?

$$\int_{0}^{1}\dfrac{1}{2x^2-x}dx$$ This is a Type II improper integral because the function is undefined at $x=\frac{1}{2}$. If I were to do this problem on my own I would split this integral in ...
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Convergence of $\int_{0}^{\infty} \frac{e^{-x}-1}{x^p} dx$

Can anybody suggest for what values of $p$, $\int_{0}^{\infty}\frac{e^{-x}-1}{x^{p}}dx$ converge ? I have tried so far ...
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Continuity of a function defined by an integral

Ok, Here's my question: Let $f(x,y)$ be defined and continuous on a $\le x \le b, c \le y\le d$, and $F(x)$ be defined >by the integral $$\int_c^d f(x,y)dy.$$ Prove that $F(x)$ is continuous on ...
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Calculate this definite integral

I get a definite integral: $\int_0^\infty x^2e^{-x^2/b}dx$ $(b>0)$. How to calculate it? Thanks in advance!
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Improper integral of odd function

I'm a student. In a recent assignment I was asked to find the mean of a Student's t multivariate distribution (which should be $\overline\mu$). I've divided the integral required to find the expected ...
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Improper Integral Convergence of Positive Continuous Function

I ask for some help or hint how to deal with this question: Suppose f(x) is continuous and positive function for all $$x\ge a$$ Prove or provide a counterexample: If $$\int_{a}^\infty f(x)dx$$ ...
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Determining whether an improper integral converges or diverges.

$$\int_{1}^{\infty}\dfrac{\sqrt{x^7+2}}{x^4}\text{dx}$$ I was told to let $f(x)=\dfrac{\sqrt{x}}{x^4}$ and $g(x)=\dfrac{\sqrt{x^7+2}}{x^4}$ then find the limit as $x$ approaches $\infty$ of ...
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improper integrals (comparison theorem)

In my assignment I have to evaluate the (improper) integral, by means of the "comparison theorem". And note whether the function is divergent or convergent. $$\int^{\infty}_{0} \frac{x}{x^3 + 1}dx$$ ...
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Improper integral of a function involving square root and absolute value.

$$\int_{-2}^{8}\dfrac{dx}{\sqrt{|2x\|}}$$ I understand that you have to split this into two integrals because at $x=0$, the function is not defined. The example showed that they split up the integral ...
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How to prove that this integral converges absolutely?

$f:[a,{\infty}[\to\mathbb{R}\$ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely ...
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Improper integral properies

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable function in every closed and bounded interval. Assuming $\int _{0}^{\infty}|f(t)|\,dt \ \$ exists. I have to prove that there exists a ...
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The integral of $\frac{\cos(x)}{x}$ from $0$ to $1$

So I have a test next week and I saw this question: proof that $\ \int_{0}^{1} \frac{\cos(x)}{x} \, \mathrm{d}x$ is not convergent. So first of all I know this is improper integral. When I saw the ...
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Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$

Compute improper integral : $\displaystyle I=\int\limits_{0}^{+\infty}\dfrac{\arctan \pi x-\arctan x}{x}dx$.
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How to prove : $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent

How to prove : If $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent then $\int\limits_{1}^{+\infty} .f(x)dx$ is too convergent.
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Does $\int_{0}^{\infty} \cos (x^2) dx$ diverge absolutely?

I believe it does, but i would like some help formulating a proof.
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Proving $\int_0^\infty\frac {1}{(1+(x\sin(5x))^2)}dx$ does not converge [duplicate]

Possible Duplicate: Why does $\int_{0}^{\infty}\frac{dx}{1+(x \sin x)^2}$ diverge? Convergence of $\int_0^\infty \frac{dx}{1+ (x^\alpha \sin x)^2}$ I understand that the following ...
Evaluating $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp(n x-\frac{x^2}{2}) \sin(2 \pi x)dx$
I want to evaluate the following integral ($n \in \mathbb{N}\setminus \{0\}$): $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp\left(n x-\frac{x^2}{2}\right) \sin(2 \pi x)dx$$ Maple and WolframAlpha ...
Evaluating $\int_{-\infty}^\infty \frac {x^2}{x^6 + 9}dx$
I am at a loss for what to do. $$\int_{-\infty}^\infty \frac {x^2}{x^6 + 9}dx$$ I tried to make $u = x^6 + 9$, $du = 6x^5$ $$\frac{1}{5}\int_{-\infty}^\infty \frac {1}{x^3u}du$$ I can rewrite $x$ ...