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.
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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\... integration improper-integrals- 4,644
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 $k , n \in \mathbb{Z}$. ... integration definite-integrals improper-integrals special-functions bessel-functions- 559
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 [... solution-verification improper-integrals monotone-functions- 379
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 ... calculus integration definite-integrals improper-integrals hyperbolic-functions- 195
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) $$\int_{0}^{1} \dfrac{\ln( 1+x)}{x}dx=\dfrac{\pi^2}{... calculus integration definite-integrals improper-integrals- 1,996
Limit with an improper integral
Is it true that $$\displaystyle\lim_{x\to 0}x\int_{0}^\infty k^5 dk=0$$ Seems like this has an indeterminate form, but I am being told it equals zero. limits improper-integrals indeterminate-forms- 1
Convergence of the integral of $x\sin(x^p)$ from $0$ to $+\infty$
Let $p\in \mathbb{R}, p>0$ for which values of $p$ does the following integral converge? $$\int_0^{\infty}x\sin(x^p)dx $$ I'm not sure how I can go about this question, I know that simply trying ... real-analysis integration analysis improper-integrals- 484
Why Does $\int_0^\infty{x^{-ln x}dx} = \sqrt[4]{e}\sqrt{\pi}$? [duplicate]
I was curious what this integral converged to, so I checked on Wolframe Alpha and it told me $\sqrt[4]{e}\sqrt{\pi}$. The problem is, I don't know why that's the answer and haven't had luck finding ... calculus integration improper-integrals- 1
Upper bound improper integral in terms of only the integrand
Suppose $f:[0,\infty)\to[0,1]$ is a nonincreasing function with $f(0)=1$, $\lim_{x\to\infty}f(x) = 0$, and also suppose that $f$ is integrable, i.e. $\int_0^\infty f(x)dx < \infty$. I am interested ... integration improper-integrals- 9
Using a definition of an improper double integral over the xy-plane as a square whose sides increase to infinity
Textbook problem (quoted): An equivalent definition of the improper integral in part (a) is $$ \iint_{\mathbb{R}^{2}} e^{-\left(x^{2}+y^{2}\right)} d A=\lim _{a \rightarrow \infty} \iint_{S_{a}} e^{-\... integration improper-integrals- 357
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: $$D = \{(x,y) \in \mathbb{R}^2 \mid x+y - a\sqrt{2} \geq 0, -x+y +a\sqrt{2} \geq 0,... integration multivariable-calculus definite-integrals improper-integrals- 79
How to prove that $\int_{0}^{\infty} e^{-\alpha \cosh(x)}\cosh(\beta x)dx<+\infty$ for every $\alpha,\beta>0$? [closed]
I'm solving another problem, and in order to be able to apply the Lebesgue dominated convergence theorem I need to study the integral $$\int_{0}^{\infty} e^{-\alpha \cosh(x)}\cosh(\beta x)dx,$$ where $... real-analysis calculus integration improper-integrals- 428
A Solution for $\int_{0}^{\frac{1}{a}} x^2e^{-ax} \: dx$ without Integration by Parts
I came across this integral while doing some physics. I'm familiar with the trick where you can solve the integrals $$\int_{0}^{\infty}x^2e^{-ax} \: dx \: \: \: \text{or} \:\int_{0}^{\infty}x^2e^{-ax^... integration definite-integrals improper-integrals- 3
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 ... real-analysis integration analysis improper-integrals gamma-function- 371
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 ... calculus integration improper-integrals- 133
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