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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-4
votes
1answer
100 views

A few fractional integral

\begin{align} & \int_0^\infty \frac{x^4}{\left( x^4+x^2+1 \right)^3}\text{d}x \\ & \int_0^\infty \frac{x^3}{\left( x^4+7x^2+1 \right)^{\frac{5}{2}}} \text{d}x \\ & \int_0^\infty ...
2
votes
1answer
346 views

A few improper integral

$$\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty ...
0
votes
1answer
125 views

Explain why $\int_0^1 dx/x$ and $\int_1^\infty dx/x$ are improper Riemann integrals

And determine whether the limits they represent exist. I evaluated both the integrals and showed that neither limits exist as finite numbers and so both integrals are divergent. I don't think I've ...
7
votes
5answers
408 views

How can I evaluate this improper integral?

I have a problem: evaluate $$\int_{0}^{\infty}\frac{\cos(x)-\cos(2x)}{x}dx\,.$$ I am told this integral is not an elementary one and that's why I am stuck where to start. Thank you for helping me.
3
votes
3answers
138 views

Calculating $\int_{0}^{\infty}dx\int_{0}^{xz}\lambda^{2}e^{-\lambda(x+y)}dy $

I wish to calculate $$\int_{0}^{\infty}dx\int_{0}^{xz}\lambda^{2}e^{-\lambda(x+y)}dy $$ I compared my result, and the result with Wolfram when setting $\lambda=3$ and I get different results. What I ...
3
votes
2answers
173 views

Calculate a double integral

I would like to ask a pretty easy question (at least I believe so). I know that: $$\phi_{11}(k) = \frac{E(k)}{4\pi k^4}(k^2 - k_1^2)$$ $$E(k) = \alpha ...
9
votes
1answer
370 views

A interesting improper integral, $ \int_{0}^1\frac{\ln x}{x^2-x-1}\text{d}x$

$$\displaystyle \int_{0}^1\frac{\ln x}{x^2-x-1}\text{d}x$$ I think there should be a smart way to evaluate this. But I cant see..
5
votes
2answers
188 views

Differentiating under the integral sign problem

Knowing that $$\int_0^\infty e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2},$$ evaluate the integral $$\int_0^\infty e^{-x^2y+1}\,dx.$$ for $y > 0$
23
votes
2answers
651 views

A nasty integral of a rational function

I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1}dx = \frac{\pi}{2}.$$ Mathematica has no ...
6
votes
4answers
394 views

Evaluate $\int_{0}^{\infty}\frac{\alpha \sin x}{\alpha^2+x^2} \mathrm{dx},\space \alpha>0$

Evaluate $$\int_{0}^{\infty}\frac{\alpha \sin x}{\alpha^2+x^2} \mathrm{dx},\space \alpha>0$$ I thought of using Feyman way, but it doesn't seem to help that much. Some hints, suggestions? Thanks.
1
vote
1answer
116 views

For what values of $a$ does this improper integral converge?

$$\text{Let}\;\; I=\int_{0}^{+\infty}{x^{\large\frac{4a}{3}}}\arctan\left(\frac{\sqrt{x}}{1+x^a}\right)\,\mathrm{d}x.$$ I need to find all $a$ such that $I$ converges.
5
votes
3answers
96 views

Prove that an integral is positive

For any positive integer $n$, consider $$\int_{0}^\infty\frac{(r^2-1)r^{n+1}}{(r^2+1)^{n+3}}dr.$$ I would like to show that it is positive. I try to write it as ...
2
votes
1answer
238 views

$\int_{0}^{\infty}{dx \over (1+x)x^\alpha} = {\pi \over \sin(\pi (1-\alpha))} $

Can you help me show that $$\int_{0}^{\infty}{dx \over (1+x)x^\alpha} = {\pi \over \sin(\pi (1-\alpha))}$$ such that $\alpha \lt1$?
-2
votes
2answers
83 views

Help me find $a$ such that the integral $I$ converges

Let $I=\displaystyle\int_{0}^{+\infty}\dfrac{dx}{x^2+x^a}$. I need to find all $a$ such that $I$ converges. Please help me!
1
vote
1answer
127 views

exercise on improper integral

$f: [a. \infty ) \rightarrow \mathbb R$ is differentiable and $\int_a^\infty f(t)dt $ and $\int_a^\infty f^{\prime}(t)dt $ are convergent then $f(t) \rightarrow 0$ as $t \rightarrow \infty$ my ...
0
votes
1answer
106 views

Improper integration [duplicate]

Possible Duplicate: How to evaluate these integrals by hand I am trying to evaluate the following: $$\int_{-\infty}^\infty \frac{\cos x}{e^x+e^{-x}}\, dx$$ using the residue theorem but I ...
1
vote
4answers
201 views

Convergence of logarithm/polynomial improper integrals

My instructor has a fondness for asking questions regarding the convergence of such integrals: $$ \int_{0}^{1} \frac{\ln(x)}{x^{1/2}}\,\mathrm dx $$ $$ \int_{0}^{1} \frac{\ln(x)}{x^{3/2}}\,\mathrm dx ...
1
vote
1answer
161 views

Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$

Prove that the improper integral $$\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$$ diverge or converge.
2
votes
0answers
99 views

Analyzing the convergence of an improper integral

I have to analyze the convergence of $$\int _{0}^{+\infty} \frac{\cos \left( x\right) -1} {x^{5 / 2}+5x^{3}}\,dx$$ I've rewritten the integral as $$ \int _{0}^{+\infty} \frac{\cos \left( x\right) ...
4
votes
1answer
161 views

Real Analysis - Conditions for Boundedness of a function

This question appeared on a Mathematics PhD Preliminary Examination - Real Analysis section. Let $Q=\{{0<x<1, 0<y<1}\}.$ For what values of $a,b$ is the function $$x^ay^b ...
5
votes
3answers
254 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/ Any help will be much appreciated.
0
votes
1answer
111 views

Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
1
vote
2answers
621 views

Fundamental solution of Heat equation

For every bounded function $u_0\in C(\mathbb R)$ the function $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{-\frac{-(x-y)^2}{4t}}u_0(y)dy$$ is a solution of $\dfrac{\partial v}{\partial ...
1
vote
0answers
161 views

comparison test for improper integrals

Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \le f\left( x \right) \le g(x)$ for all $x \in \left( {a,b} \right)$; $a$ can be $ - \infty $ and $b$ can be $ + \infty $. Prove: ...
1
vote
0answers
54 views

how to calculate $\int_{[0,1]}x^{-\alpha}dx$

$0 <\alpha <1$,prove $x^{-\alpha}\in L([0,1])$,and calculate $$\int_{[0,1]}x^{-\alpha}dx$$ Here is my method. We only need to show that $\int_{[0,1]}x^{-\alpha}dx \leq \infty$. ...
1
vote
1answer
107 views

Prove $\lim_{n \to \infty} \int_0^1f(x)e^{inx^3}dx = 0$

Assume that $f:[0,1] \to R$ is a smooth function. Prove that $$\lim_{n \to \infty} \int_0^1f(x)e^{inx^3}dx = 0.$$ Attempt at solution: I think the solution may require the interchange of limit and ...
2
votes
1answer
164 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
3
votes
2answers
145 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
0
votes
2answers
81 views

Bounding the integral $ \int_1^\infty \frac{ (\log(y))^n }{y^2} \ dy $

I'm trying to show that the integral $$ \int_1^\infty \frac{ (\log(y))^n }{y^2} \ dy $$ is convergent for every real number $ n \geq 1$. If $ n < 2$, I can bound $ |\log(y)|$ by $y$ and hence show ...
-3
votes
2answers
77 views

Improper Riemann integrals

Calculate the improper Riemann integral $$I_a := \int\limits_0^1\frac{dx}{x^a}$$ Does $I_a$ have a limit as $a \to 1^-$?
2
votes
1answer
207 views

Show that $\int\limits_{-\infty}^{\infty} \! \mathbb{e}^{-x^2} \, \mathrm{d}x$ = $\sqrt{\pi}$. [duplicate]

Possible Duplicate: Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$ Show that $\int\limits_{-\infty}^{\infty} \! \mathbb{e}^{-x^2} \, \mathrm{d}x$ = $\sqrt{\pi}$. ...
3
votes
4answers
135 views

Divergence of a simple improper integral

I have a reduced a problem I have been working on to showing that $$\int_{0}^{\infty} \ln \left(\frac{x+2}{x+1} \right)dx$$ diverges, but I'm not sure how to show this. Would anyone be able to help me ...
2
votes
0answers
51 views

Whether $\lim_{x \to +\infty} f(x) = 0 $ when improper integral is convergent? [duplicate]

Possible Duplicate: Prove: $f(x)$ continuous function and $\int_a^\infty |f(x)|\;dx \lt \infty$ so $\lim_{ x \to \infty } f(x)=0$ Assume $f(x)$ positive and continuous,for improper ...
4
votes
3answers
337 views

Integrating using Laplace Transforms

$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$ I'm supposed to solve this using Laplace Transformations. I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
0
votes
3answers
119 views

Find a $f(x) \not=0$ satisfies $\int_1^{\infty}(1-\frac{1}{x})f(x)dx=0$

can we find a function $f(x)\not=0$,such that $$\int_1^{\infty}\left(1-\frac{1}{x}\right)f(x)dx=0$$ who can give an instance ? thanks
1
vote
3answers
316 views

Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem

I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$. But how to evalulate ...
1
vote
4answers
69 views

Convergence of an improper integral - II

I'm not able to find the value of:$$ \int_a^\infty \frac{1}{x^2+1}dx, a>0 $$ What I can do?
0
votes
2answers
60 views

Convergence of an improper integral - I

What's the value of:$$ \int_a^\infty x^{-2}dx, a>0 $$ And why it converge?
0
votes
3answers
76 views

Why/How does this function become 1 when integrated?

Question asks to show that if $$f(x)= \begin{cases} \frac14xe^\frac{-x}{2} & x>0\\[8pt] 0 & \text{elsewhere}, \end{cases}$$ then $$\int_0^\infty f(x)\,dx=1.$$ I get $$\int f(x) = \frac14 ...
2
votes
2answers
226 views

Change of variables in improper double integral - how to find the limits?

Consider the following integral (originating from the product of the Laplace transforms of $f$ and $g$): $$\int_0^\infty \int_0^\infty f(u)\ g(v) e^{-s(u+v)}\ du\ dv.$$ For this integral, the ...
2
votes
0answers
172 views

Closed form of the series ,$\sum_{k=0}^{\infty}\frac{(-1)^k k!}{(k+1)^{(k+1)}}x^k$

$x,y>0$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{xt+e^{y t}} dt$$ if $x=0$ then $f(0,y)=1/y$ $$f(x,y)=\int_{0}^{\infty} \frac{1}{e^{y t}(1+xte^{-y t})} dt=\int_{0}^{\infty} \frac{e^{-y ...
3
votes
1answer
146 views

integral evaluation of an exponential

let be the function $$ e^{-a|x|^{b}} $$ with $ a,b $ positive numbers bigger than zero then how could i evaluate this 2 integrals ? $$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$ here 'c' can ...
1
vote
3answers
98 views

How to evaluate $\int_{-\infty}^{\infty}(1+kx^2)^{-2}dx$

Can someone give a hint to evaluate the following integral? $$\int_{-\infty}^{\infty}(1+kx^2)^{-2}dx$$ where $k>0$.
5
votes
2answers
251 views

integral of exponential divided by polynomial

I would like to solve the integral $$A\int_{-\infty}^\infty\frac{e^{-ipx/h}}{x^2+a^2}dx$$ where h and a are positive constants. Mathematica gives the solution as $\frac\pi{a}e^{-|p|a/h}$, but I have ...
1
vote
1answer
630 views

improper Riemann integral and Lebesgue integral

Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral $\lim_{\epsilon \to 0} ...
9
votes
2answers
327 views

$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \mathrm dx$ Evaluate Integral

Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$
2
votes
2answers
801 views

Convergence/absolute convergence of $\int_0^\infty \frac{\cos x}{1+x}dx$ (Baby Rudin P6.9)

Problem 6.9 of Rudin's PMA asks the reader to demonstrate conditions in which indefinite integrals that satisfy the definition $$\int_a^\infty f(x)dx = \lim_{b\to\infty} \int_a^b f(x)dx$$ can ...
1
vote
2answers
357 views

Evaluating an improper integral with radical in denominator

For $a,b\in\mathbb{R}$ and $a<b$, I'd like to evaluate the following improper integral $$ \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} $$ Since I'd like to evaluate the integral rather than just make sure ...
1
vote
3answers
168 views

Show an improper integral converges

I would like to show that the following improper integral converges, but it's been a while since I've done this sort of calculus and I'm drawing a blank: $$ \int_0^1 \frac{dx}{\sqrt{x^3-x}} $$ My ...
10
votes
4answers
330 views

How to evaluate this integral $\int_{-\infty}^{+\infty}\frac{x^2e^x}{(1+e^x)^2}dx$?

I need to evaluate $$\int_{-\infty}^{+\infty}\frac{x^2e^x}{(1+e^x)^2}dx$$ I think the answer is $\frac{\pi^2}{3}$, but I'm not able to calculate it.