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 ∞ or −∞, or as both endpoints approach limits.

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3
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0answers
86 views

Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
3
votes
1answer
34 views

Probability Density Function Equation, Multivariable Calculus

I have the following problem: The formula for the normal distribution has a π in it. In this simplified version of the normal probability density function, solve for C. The correct answer has π in ...
0
votes
3answers
79 views

Calculate $\int_{-\infty}^{\infty} e^{-x^2}\cos (ax) dx$ using Taylor series cosine

Let $a > 0$. Im trying to show that $\int_{-\infty}^{\infty} e^{-x^2} \cos (ax) dx = \sqrt{\pi}e^{-\frac{1}{4}a^2}$. I'm taking a course on measure theory, and I want to prove this using the ...
0
votes
0answers
12 views

Decay of a Fourier Transform with parameter

Given $t>0$, consider the following function $f_t:\mathbb{R}\rightarrow\mathbb{C}$ $$f_t(x)=\begin{cases}e^{-tx-itx^2}&x\geq 0\\ 0&x<0\end{cases}$$ Now, let $\widehat{f_t}$ the Fourier ...
0
votes
1answer
40 views

Testing the convergence of an improper integral.

Find all the real values of $p$ and $q$ so that $$ \int_{0}^{1} x^{p}\biggl(\ln\frac{1}{x}\biggl)^q dx$$ converges. I tried using comparison test but couldn't solve it. Please help me.
3
votes
1answer
39 views

How do I express $\int_0^{\frac{\pi}{2}}\sin^{2m-1}\left(t\right)\cos^{2n-1}(t)dt$ using the Gamma-function?

We define the Gamma function as: $$\Gamma(p)=\int_0^{+\infty}e^{-x}x^{p-1}dx$$ I was advised to rewrite the integral as $\sin(t)^{2m-1}\cos(t)^{2n-2} d \sin(t)$, and substitute $ t = \sin(t)$ which ...
0
votes
1answer
58 views

How can I calculate $\int\limits_{0}^{\infty}e^{-e^x}dx$?

I have no idea to start this. Any hint or solution would be highly appreciated.
2
votes
2answers
79 views

Is $\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{1 - \cos x\cos y\cos z}\, dx\, dy\, dz$ finite?

Is $$\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{1 - \cos x\cos y\cos z}\, dx\, dy\, dz$$ finite? Mathematica is throwing me an error, but I suspect the integral converges, since the ...
2
votes
2answers
29 views

For $p>0$, does $\int_1^\infty x^{-p/x}$ diverge?

For $p>0$, does $\int_1^\infty x^{-p/x}$ diverge? I've tried the root test, the comparison test, and the limit comparison test without success. Any assistance would be appreciated.
0
votes
2answers
45 views

Convergence of $\int_{-1}^{1}\frac{x-1}{x^{5/3}}dx$

Find whether the following integral converges or diverges.$$\int_{-1}^{1}\frac{x-1}{x^{5/3}}dx$$ Attempt- I tried breaking the integral into two parts- one from -1 to 0 and other from 0 to 1 and then ...
0
votes
1answer
97 views

Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour

I need to try to evaluate $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ and it seems like this is supposed to be done using some sort of rectangular contour based on looking at other questions. My ...
0
votes
1answer
49 views

Integral of power series over a region

Let $$ F(x) = \sum_{n=0}^ \infty a_nx^n$$ where the power series converges in a neighborhood of the origin. Compute $$ \mu(F)= \sup \{ \delta > 0 : \text{there exist} \ \epsilon > 0 \ ...
0
votes
1answer
31 views

Transformation of an improper integral of the square root of a rational function

Why is this true? $$\int_{-\infty}^{\infty}\sqrt{\frac{1}{(t^2-1)^2}-\frac{(n+1)t^{2n}}{(t^{2n+2}-1)^2}}dt=4\int_{0}^{1}\sqrt{\frac{1}{(t^2-1)^2}-\frac{(n+1)t^{2n}}{(t^{2n+2}-1)^2}}dt$$ I know that: ...
1
vote
4answers
50 views

Calculation of $\frac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\exp\left(-\frac{u^2}{2\sigma{^2}(x)}\right)\ du$

How to show that $$\dfrac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\operatorname{exp}\left(-\dfrac{u^2}{2\sigma{^2}(x)}\right)\mathop{du}=\sqrt{\dfrac{2}{\pi}}\sigma(x)$$ I think it has to ...
1
vote
1answer
43 views

Improper integral $\int_{-\infty}^{+\infty}\frac{1}{a x^2 + bx + c} dx$, $a > 0,a x^2 + bx + c>0$

I have a question about improper integral. If you can help me , I appreciate that. If a > 0 and the graph of $y=a x^2 + bx + c$ lies entirely above the x-axis, show that $$ \int_{-\infty}^{+\infty} ...
2
votes
1answer
22 views

Improper integral $\int_0^3 (1-x^{2}\sin(\frac{1}{x^{2}})) dx$

I have to determine the convergence of $\int_0^3 (1-x^{2}\sin(\frac{1}{x^{2}})) dx$ Can I say that $(1-x^{2}\sin(\frac{1}{x^{2}}))\leq1+x^2$ and since $\int_0^3 1+x^2 dx$ is not even improper, ...
2
votes
2answers
75 views

Convergence of improper integral $\int_0^\infty x^{\alpha +1} e^{-x} dx$

Determine the convergence of the following improper integral as $\alpha \in \mathbb{R}$ varies: $\int_0^\infty x^{\alpha +1} e^{-x} dx$ I tried to do it in this way $e^{-x}<\frac{1}{x^{\beta}}$ ...
9
votes
5answers
401 views

Evaluating $\int_0^{\infty} \frac {e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$

I am trying to evaluate this integral $$I=\int_0^{\infty} \frac {e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$$ for $a \in \mathbb R_{>0}$. Any ideas? In the case $a=\pi$ we have $I= F - e$ where $F$ ...
0
votes
1answer
26 views

Reversing the direction of a contour integral

If $$\int_{C} f(z) dz$$ is some contour integral over a closed curve $C$, and $-C$ is the contour taken in the opposite direction, can $$ \int_{-C} f(z) dz$$ be treated as a closed curve around the ...
1
vote
1answer
50 views

If $\lim\limits _{x\to\infty}\left(\log f\right)'\left(x\right)$ is negative, then $\int_{0}^{\infty}f\left(x\right)dx $ converges?

Let $f:[0,\infty)\to\mathbb{R}$ be a positive differentiable function, and suppose that $\lim\limits _{x\to\infty}\left(\log f\right)'\left(x\right)$ exists is negative. Prove that ...
0
votes
2answers
49 views

Help analyzing the convergence of $\int_{2}^{+\infty} \frac{1}{\ln^p(x) x^s}\,dx$

so far I've been able to establish what happens when, $p=0$ and $s=0$, diverges $p=0$ and $s>1$, diverges $p=0$ and $s<1$, converges $p=0$ and $s=1$, diverges But when $p\neq 0$ and $s\in ...
2
votes
1answer
25 views

Integral evaluation by sandwiching

Evaluate $\lim_{p\to 0+}\int_0^p t^{1+t} \mathrm{d}t$. Trying to integrate this is a pipe dream, so my hope is to somehow achieve the following scenario: $$L =\lim_{p\to 0+}\int_0^p ...
3
votes
2answers
58 views

How to compute $\int_{-\infty}^{+\infty} 2^{-4^t}(1-2^{-4^t})\,dt=\frac{1}{2}.$

How to compute $$\int_{-\infty}^{+\infty} 2^{-4^t}(1-2^{-4^t})\,dt=\frac{1}{2}.$$ I'm interested in more ways of computing this integral. My thoughts : Let $y=4^t$ we got ...
0
votes
1answer
52 views

Test the convergence of the integral $\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$

Test the convergence of the following integral $$\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$$I can not find the indefinite integral of the integrand so that we can check at the limits $-\infty$ and ...
1
vote
1answer
58 views

Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $

I'm trying to figure out this integral but cannot figure out the right substitution $$ \int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}dx $$
5
votes
0answers
88 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
0
votes
1answer
22 views

General technique to check the convergence of an improper integral?

Which of these integrals converge ? I am confused about how to check for the convergence when the functions are more complex inside the integral. My attempt: in option C : integrating gives -2 and ...
0
votes
2answers
53 views

Find $k$ such that $\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$

I'm trying to solve for $k$ given that the integral $$\int_0^{\infty} ky^3 e^{\frac{-y}{2}}dy = 1.$$ I can see that I can pull out k to get $$k \int_0^{\infty} y^3 e^{\frac{-y}{2}}dy = 1.$$ However, ...
3
votes
3answers
80 views

Evaluation of $ \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}dx$

$\bf{My\; Try:}$ Let $$\displaystyle I = \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}\,dx = \underbrace{\int_{0}^{1}\frac{x\ln x}{(1+x^2)^2}\,dx}_{I_{1}}+\underbrace{\int_{1}^{\infty}\frac{x\ln ...
1
vote
1answer
31 views

Does $\int^3_0 (9-x^2)^{-3/2}\, dx$ converge?

Does $$\int^3_0 {dx \over (9-x^2)^{3 \over 2}}$$ converge? I tried to compare to $1 \over x^3$ and $1 \over x^{3 \over 2}$ using compare test and limit compare test but it didn't work out. (I ...
0
votes
0answers
12 views

Proof of convergence theorem for extended Riemann Integral.

The theorem states: Suppose that f is continuous on $[a,b[$ and that $f(b^-) = + \infty$. Say $\alpha > 0$ and $\lim_{x\to b} (b-x)^\alpha f(x) = K_\alpha, K_\alpha \in \mathbb{R}\cup \{+\infty ...
0
votes
0answers
20 views

Mapping of infinite domain

I am integrating an ODE. For simplicity let us consider a simple one: $\dot{x} = u$ The solution is simply, $x = ut + x_o$. Now suppose my $x$ goes from 0 to $\infty$. But I want to do the ...
5
votes
1answer
79 views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

I am trying to evaluate $$ \int_0^1 \frac{\ln(1+x)}{x}\,dx $$ I started by using the Taylor series for $\ln (1+x)$ $$\begin{align*} \int_0^1 \frac{\ln(1+x)}{x}\,dx &= ...
0
votes
0answers
17 views

Surjectivity of a linear function on $C^\infty$ vector space

Given $$T:f \in E \longmapsto \left(x \mapsto \int_0^{+\infty}\frac{f(t)}{1+t^2+x}\right) \in C^\infty(]-1, +\infty[)$$ Where $E$ is the vector space of function whom the previously defined integral ...
2
votes
1answer
61 views

Proving $\int_{0}^{\infty} e^{-x^2}dx=\sqrt{n}\int_{0}^{\infty} e^{-nx^2}dx$

Prove: $\int_{0}^{\infty} e^{-x^2}dx=\sqrt{n}\int_{0}^{\infty} e^{-nx^2}dx$ for every natural number I tried a lot of things, I think induction is the way to go here but I couldn't really get ...
3
votes
2answers
92 views

Computation of integral

Consider the following integral: $$ I = \int_{0}^{\infty}{\int_{0}^{\infty}{x e^{-\left(2-\frac{1}{b^2}\right) x^2} e^{-\frac{1}{2}(y + a x)^2}} dy dx}$$ I was able to integrate over $y$ to get: $$ ...
3
votes
1answer
49 views

Correctness of the integral $\int_a^bf(x)\,dx$ when $f$ is not defined at $a$ and $b$

Knowing that the function $$\frac{1}{\sqrt{1-x^2}}\ $$ is defined only for $-1 < x <1$, are the limits of integration below allowed? $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}} \,dx= \pi$$ PS: As ...
4
votes
2answers
151 views

Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$

How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}dx=\frac{\pi}{2e}$$ I've tried several basic approaches like substitution and IBP but can't move forward.
1
vote
0answers
37 views

How to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0\Rightarrow\lim\limits_{x\rightarrow\infty}f'(x)$? [duplicate]

From a exercise list: Let $f:[a,+\infty)\rightarrow \mathbb{R}$ with continuous second derivative and such that $\lim\limits_{x\rightarrow\infty}f(x)=0$ and ...
2
votes
1answer
81 views

How to Solve a differential equation with both x and y?

Solve $\dfrac{dy}{dx}=\dfrac{y-3}{y^2+x^2}$ given that it passes through $(0,1)$. Right now I do not yet know how to solve differential equations with both $x$ and $y$ that you cannot separate. ...
1
vote
2answers
64 views

Calculating integrals with asymptotes?

Find $\displaystyle\int^2_0 \dfrac{1}{(1-x)^2} dx$. Is there a way of doing this without considering the asymptote at $x=1$? What if you didn't know at first that there was indeed an asymptote at ...
1
vote
1answer
23 views

Improper integral for $\ f \rightarrow \infty$

Maybe it's a stupid question but i just want to be sure of that. If a function approaches $\infty$ as $\mathrm x \rightarrow \infty$ it is useless trying to evaluate its improper integral from (for ...
1
vote
0answers
33 views

Improper integral:- Change of Variable.

$$ I=\int _{ -\infty}^{ \infty} \frac{\log(\lvert t\rvert)}{x^4+t^2} \, dt $$ ATTEMPT:- Since the function is even, the following property may be used: $$\int_{-a}^a f(x) \, dx=2\int_0^a f(x) \, dx, ...
0
votes
0answers
19 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
3
votes
2answers
39 views

Prove $\int_{0}^\infty \mathrm{d}y\int_{0}^\infty \sin(x^2+y^2)\mathrm{d}x=\int_{0}^\infty \mathrm{d}x\int_{0}^\infty \sin(x^2+y^2)\mathrm{d}y=\pi/4$

How can we prove that \begin{aligned} &\int_{0}^\infty \mathrm{d}y\int_{0}^\infty \sin(x^2+y^2)\mathrm{d}x\\ =&\int_{0}^\infty \mathrm{d}x\int_{0}^\infty ...
1
vote
0answers
29 views

Is $\max \int_{- \infty}^{\infty} \frac{dx}{\pi (1 + x^2 + f ' (x)^2 )} $ a uniqueness condition here?

Let f(x) be a real-differentiable function with $f′(x)>0,f′′(x)>0 $ and $$ f(f(x)) = \exp(x) $$ for all real $x$. Tommy1729 adds the optimization condition $$ max \int_{- \infty}^{\infty} ...
2
votes
2answers
40 views

Improper integral

$\int_{-1}^{1}\frac{e^{x}}{1+x}dx$ Anyone know how to solve this? I have tried to find a substitute, or a function to compare with, but I don't have any clue whatsoever what to do. Wondered if ...
0
votes
1answer
24 views

Product of improper integrable riemann function and integrable function.

I have the following problem while working with linear differential equations: I'm told to proof that the following system: $t^{-\sigma}x' = A(t) x$ where $A: \mathbb{R} \to {M_N(\mathbb R)}$ is ...
1
vote
2answers
64 views

Is $\int \limits _{0}^{1}x^\alpha|ln(x)|^\beta dx$ convergent? [closed]

Is $\int \limits _{0}^{1}x^\alpha|ln(x)|^\beta dx$ convergent, where $\alpha >0, \beta >0$? Thanks .
0
votes
1answer
45 views

$f:[a,\infty)\to\mathbb R$ be continuous such that $\int_a^\infty f(x)dx$ exists finitely , then $\lim_{x \to \infty}f(x)=0$ ? [duplicate]

Let $f:[a,\infty)\to\mathbb R$ be a continuous function such that $\int_a^\infty f(x)dx$ exists finitely , then is it true that $\lim_{x \to \infty}f(x)$ exists and is equal to $0$ ? If not , then ...