![Questions Tagged [Improper-Integrals]](https://docuprint.cloud68.co/uploads/questions-tagged-improper-integrals_1200.jpg)
Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.
7,051
questions
2
votes
1
answer
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Problem with the integral $\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t$
I'd like to prove, using a partial fraction decomposition (I don't want to use residue calculus), that
$$\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t=\frac{\pi}{2n\sin\frac{m\pi}{2n}}$$
where $1\...
0
votes
0
answers
25
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Integral of product of modified Bessel functions: $ \int_0^{\infty} r e^{- a r^2} I_k (b r^2) I_{2k-n} (c r) d r $
The following integral appeared in my research recently.
$$ \int_0^{\infty} r e^{- a r^2} I_k (b r^2) I_{2k-n} (c r) d r , \tag{*} $$
where $c \geq 0$, $a > b \geq 0$, and $k , n \in \mathbb{Z}$. ...
0
votes
0
answers
30
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If $f$ is strictly increasing and integrable on $[0,+\infty)$ then $f(x)<0$ for any $x \in [0,+\infty)$
Prove that if $f$ is strictly increasing and integrable on
$[0,+\infty)$, then $f(x)<0$ for any $x \in [0,+\infty)$.
I tried this: assume by contradiction that there exists at least one $x_0 \in [...
4
votes
2
answers
135
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Definite integral $ \int _{0}^{\infty } x\cdotp \tanh( 2x) \cdotp \ln(\coth x)\mathrm{d} x$
I want to show that
$\displaystyle \int\limits _{0}^{\infty } x\cdotp \tanh( 2x) \cdotp \ln(\coth x)\mathrm{d} x=\frac{\pi ^{2} \cdotp \ln( 2)}{2^{4}}\tag*{}$
I tried integration by parts, Feynman ...
1
vote
2
answers
152
views
Let $\lim_{n\to \infty} \sum_{r=1}^n \frac{1}{r^2}=\frac{\pi^2}{6},$ then which of the following is/are true?
For any natural number $n,\;$ Let $$\lim_{n\to \infty} \sum_{r=1}^n \dfrac{1}{r^2}=\dfrac{\pi^2}{6},$$ Then which of the following is/are true?
(A) $$\int_{0}^{1} \dfrac{\ln(
1+x)}{x}dx=\dfrac{\pi^2}{...
0
votes
0
answers
30
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Limit with an improper integral
Is it true that
$$\displaystyle\lim_{x\to 0}x\int_{0}^\infty k^5 dk=0$$
Seems like this has an indeterminate form, but I am being told it equals zero.
0
votes
1
answer
55
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Convergence of the integral of $x\sin(x^p)$ from $0$ to $+\infty$
Let $p\in \mathbb{R}, p>0$ for which values of $p$ does the following integral converge?
$$\int_0^{\infty}x\sin(x^p)dx $$
I'm not sure how I can go about this question, I know that simply trying ...
-2
votes
0
answers
38
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Why Does $\int_0^\infty{x^{-ln x}dx} = \sqrt[4]{e}\sqrt{\pi}$? [duplicate]
I was curious what this integral converged to, so I checked on Wolframe Alpha and it told me $\sqrt[4]{e}\sqrt{\pi}$. The problem is, I don't know why that's the answer and haven't had luck finding ...
0
votes
0
answers
41
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Upper bound improper integral in terms of only the integrand
Suppose $f:[0,\infty)\to[0,1]$ is a nonincreasing function with $f(0)=1$, $\lim_{x\to\infty}f(x) = 0$, and also suppose that $f$ is integrable, i.e. $\int_0^\infty f(x)dx < \infty$. I am interested ...
0
votes
1
answer
20
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Using a definition of an improper double integral over the xy-plane as a square whose sides increase to infinity
Textbook problem (quoted): An equivalent definition of the improper integral in
part (a) is
$$
\iint_{\mathbb{R}^{2}} e^{-\left(x^{2}+y^{2}\right)} d A=\lim _{a \rightarrow \infty} \iint_{S_{a}} e^{-\...
3
votes
1
answer
80
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Calculate double integral $\iint_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}dxdy$ over unbounded region $D$
Calculate
$$I = \iint\limits_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}\,dx\,dy$$
where the region of integration is:
$$D = \{(x,y) \in \mathbb{R}^2 \mid x+y - a\sqrt{2} \geq 0, -x+y +a\sqrt{2} \geq 0,...
0
votes
0
answers
68
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How to prove that $\int_{0}^{\infty} e^{-\alpha \cosh(x)}\cosh(\beta x)dx<+\infty$ for every $\alpha,\beta>0$? [closed]
I'm solving another problem, and in order to be able to apply the Lebesgue dominated convergence theorem I need to study the integral
$$\int_{0}^{\infty} e^{-\alpha \cosh(x)}\cosh(\beta x)dx,$$
where $...
0
votes
1
answer
84
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A Solution for $\int_{0}^{\frac{1}{a}} x^2e^{-ax} \: dx$ without Integration by Parts
I came across this integral while doing some physics. I'm familiar with the trick where you can solve the integrals
$$\int_{0}^{\infty}x^2e^{-ax} \: dx \: \: \: \text{or} \:\int_{0}^{\infty}x^2e^{-ax^...
2
votes
0
answers
64
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Let $a>0$. Prove the improper integral $\int_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx$ converges for $k>2$
There is a hint which says $\left|\int\limits_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx\right|\le C a^{k-2}$ where C is some constant.
I somehow feel that I need to ...
0
votes
1
answer
38
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How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers?
How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers?
I have tested this for specific numbers and it looks like we ...
