Improper integral Definition and 238 Threads

Homework Statement Let f be a continuous function on [1,∞) such that \lim_{x\rightarrow ∞}f(x)=α. Show that if the integral \int^{∞}_{1} f(x)dx converges, then α must be 0. Homework Equations Definition of an Improper Integral Let f be a continuous function on an interval [a,∞). then we...

Determine the convergence or divergence of this following improper integral: \int_2^∞ \frac {1}{(x^3+7)^{\frac{1}{3}}} So I'm trying to find something easy to compare this to, any help?

Homework Statement Consider the function f(x)=1/(x^p). When p>1, the integral of 1/(x^p) from 1 to infinity converges. i) For what values of p does the integral of 1/(x^p) from 0 to 1 converge? (0<p<infinity, p does not equal 1). ii) Confirm the answer by re-writing the...

Homework Statement Consider the function f(x)=1/(x^p). When p>1, the integral of 1/(x^p) from 1 to infinity converges. i) For what values of p does the integral of 1/(x^p) from 0 to 1 converge? (0<p<infinity, p does not equal 1). ii) Confirm the answer by re-writing the integral...

What is the improper integral of sin[x]/(1+x^2) from -infinity to infinity equal to? It doesn't seem to be integrable in the real number system. Is it 0, because sin[x] is an odd function and 1+x^2 is an even function, and an odd function divided by an even function is equal to an odd...

Hello.. Wondering whether I am right that \int_{-1}^{1} \frac{dx}{x} = 0, and therefore it's convergent, because my teacher insists on splitting into two divergent improper integrals then he says it's divergent. Thanks in advnace,,

Is ∫eiωtdω = √(2\pi) (from -∞ to ∞) It should be so if my books derivation of the Fourier transform is correct, but they don't really explain why the above equality is correct? can anyone show me how they get to that result for that integral or does anyone perhaps have a good link?

Find upper limit of improper integral -- Numerical Integration Hi, I have the following complicated integral willing to integrate numerically. The integral is: \int _0^{\infty }x\frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\beta/x}dx . We know that the integral converges to...

EDIT: I am sorry if you can't understand the integral straight away, I am not familiar with using the notion provided by this forum. I tried, but... Homework Statement Find all values of the constant α for which the integral: ∫ [(x/(x^2 + 1)) - (3a/(3x + 1))] dx (from 0 to +infinitity)...

When the following improper integral converges? When it diverges? $$ \int^{\infty}_{0} \frac{x^{\alpha}dx}{1+x^{\beta}\sin^2(x)} $$

My first post on the new forums is going to be a challenge problem.$\displaystyle \int_{0}^{\infty} \frac{\sin ax \ \sin bx}{x^{2}} \ dx \ , \ a > b \ge 0$

Homework Statement Suppose f is real-valued, bounded, continuous, and non-negative and suppose \int_x^\infty f(t)\,dt is convergent (is finite) for all x. Is it true that \lim_{x\rightarrow \infty} {\int_x^\infty f(t)\,dt} = 0 \ ? Homework Equations The Attempt at a Solution I can't...

Homework Statement The problem is ∫e^x/(e^x)-1 to be evaluated from -1 to 1. Homework Equations The Attempt at a Solution I got the integral as ln|(e^x)-1| So, for the first part, evaluating from -1 to 0, with t being the limit at 0 I got this: ln|(e^t)-1| -...

Hi everybody I was trying to prove that \int_{-\infty}^{\infty}e^{\imath (k - k') x}dx = 2\pi\delta(k-k') by solving \lim_{L\rightarrow \infty} \int_{-L}^{L}e^{\imath (k - k') x}dx knowing that \delta(x)=\lim_{g\rightarrow \infty}\frac{\sin(gx)}{\pi x} But is there a way of proving this...

Homework Statement (a) we define the improper integral (over the entire plane R2) I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA where Da is the disk with radius a and center the...

Homework Statement ∫(dx/((1+x^2)^2) from 0 to ∞ Determine whether the improper integral converges and if so, evaluate it.Homework Equations 1+ tan^2(x) = sec^2(x) 1/sec(x) = cos(x) The Attempt at a Solution Initially I had no idea how to approach this problem. The answer in the back of the...

Homework Statement ∫^{∞}_{1}\frac{dx}{x(2x+5)}Homework Equations Mathematica told me that the answer was \frac{1}{5}nl(\frac{7}{2}) The Attempt at a Solution This is my work. ∫^{∞}_{1}\frac{dx}{x(2x+5)} = lim_{M\rightarrow∞}∫^{M}_{1}\frac{dx}{x(2x+5)} =...

Hello everyone! My first time on this forum :). Could anybody please help me with the following? I need to know whether this: ∫abs(2x-1)dx is an improper integral and if so, why? I use abs( as a notation for absolute value. Thanks for the help!

\int_{0}^{2} \frac{1}{1-x^{1/3}} dx I then would break up into: \int_{0}^{1} \frac{1}{1-x^{1/3}} dx + \int_{1}^{2} \frac{1}{1-x^{1/3}} dx I'm lost on where to go from here. The integral looks so simple but I'm not sure if I should make a u sub or a trig sub. Could someone give me a hint...

Homework Statement Prove or disprove: b\int_b^∞ f(x) dx ≤ \int_b^∞ xf(x) dx for any b≥0 and f(x)≥0Homework Equations N/A The Attempt at a Solution Ok this question has caused me quite some problems. I have come to the conclusion that this needs to be proven rather than disproven...

Homework Statement find the integral from 1 to infinity of (arctanx/x^2)dx Homework Equations The Attempt at a Solution i used integration by parts: u=arctanx du=1/(1+x^2)dx dv=x^-2dx u=(-1/x) -arctanx/x + [(1/(x)(1+x^2))dx]from 1 to infinity i have a partial solution in...

How to show this? \int_{-\infty}^{+\infty}e^{-i2\pi xs}ds=\delta(x) This is a part of a problem of "Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 100-101, 1999". This isn't a homework, I found it...

Hi this is very complicated integral i couldn't solve can you help me ? how does it solve

Been doing some calculus review to knock the rust off for this coming fall semester and I got stuck... Homework Statement From Stewart's book (Early Transcendentals: 6E): (7.8 pg517 #69) Determine how large the number "a" has to be so that: \int\stackrel{\infty}{a}\frac{1}{x^{2}+1}dx...

Homework Statement \int \frac{arctan(x)}{2+e^x}dx where the interval of the integrand is from 0 to infinity. In order to use the comparison method I need to compare 2 functions but I am having so much difficulty figuring out what function to compare it to. Its not just this particular...

Homework Statement the integral of ∫dx/(x^a 〖(lnx)〗^b ) from zero to infinite Homework Equations The Attempt at a Solution i divided the equation to two parts I and J where I is from 0 to 1 and J from 1 to infinite then i tried to to neighbourhood of zero i couldn't know how i...

Homework Statement hi, I've been working on this for the past half hour and i can't seem to get the right answer definite integral from 1 to infinity, dx/[(x)(2x+5)]...

Decide if the following improper integral converges or not Integrate 1/x2 limits are 1 0 Now by simple integration we can see that: [x-2] = -x-1 Substituting in 1 and 0: -(1/1) - 0 So series converges to -1 Is this correct?

Homework Statement For what values of K is the following integral improper? \int\stackrel{K}{0}x^2 / (x^2-19x+90) dxI'm stuck on this question. I understand mechanically, that the integration require partial fraction decomp, which results in -9ln(x-9) (from 0 to K) + 10ln(x-10) (from 0 to...

I would like to solve the following integral but I am unsure of the best way to solve it: \int_{0}^{H}xsin(\frac{w}{x})cos(\frac{x}{w})cosh(\frac{H}{w})dx Is it possible to use integration by parts?? Thanks in advance

Homework Statement For what values of r does \int(from 0 to infinity) xre-x dx converge? I assume that the problem refers to r as any real number. 2. The attempt at a solution I have given this a try but I am really not confident that I did it right... First i used integration...

Homework Statement Find if the improper integral converges or diverges; If converges find its exact value. Integral from 0 to infinity of (Xe-x)dX Homework Equations The Attempt at a Solution Using limit properties, i found that the integral was zero. Does it make sense that the...

Homework Statement Does the integral from 1 to infinity of ([(Cos[Pi x])^(2x)]/x)dx converge? Homework Equations N/A The Attempt at a Solution I claim it converges (based on how small the values of the function get when x is not an integer), but I'm not really sure how to...

Homework Statement Show that ∫sin(ax) / xp dx interval: [0, ∞] converges if 0 < p < 2. Homework Equations This is the chapter where we learn that ∫f(x)g(x)dx converges if ∫f(x)dx is bounded and g'(x) is continuous, g'(x) < 0, and g(x) --> 0. The Attempt at a Solution...

Homework Statement Gamma function is defined for all x>0 by rule \Gamma(x)=\int0\inftytx-1e-tdt Find a simple expression for \Gamma(n) for positive integers n. Answer is \Gamma(n)=(n-1)! Homework Equations The Attempt at a Solution...

Homework Statement \int\limits^{ +\infty }_{0}\frac{ \sqrt{x} \mbox{d} x }{ x^2+1 } The Attempt at a Solution I make substitution \sqrt{x}=t and then \int^{ +\infty }_{0}\frac{ t^2 \mbox{d} t }{ t^4+1 } and now this function is odd, so I make a half circle and count residues yeah?

Hi, I came across this interesting integral \int_{-\infty}^{\infty}\frac{d}{dx}(\arctan x) dx=\pi I can derive the solution analytically, but I cannot think of a geometric proof. Does anyone know geometrically why this integral is equal to pi?

Homework Statement integral: \int\limits_0^\infty\frac{\mbox{d}x}{\left(x^2+1\right)\left(x^2+4\right)} The Attempt at a Solution normally i would do I=\frac12\int\limits_{-\infty}^\infty\frac{\mbox{d}x}{\left(x^2+1\right)\left(x^2+4\right)} and now count residues but is there any other...

Hello i stumbled over a question and I'm not sure how to proof/ solve hat: At how many points a function can not be defined to be nevertheless integrable (improper integral)? Thx in advance!

Homework Statement Find the value of Integral from 2 to infinity of 1/(4x-2)^5 dx. Homework Equations The Attempt at a Solution When I integrated I came up with -(1/(16(2-4x)^4)), top - bottom which ends up being 0-(-.0004822) but it says that is wrong. Help please!

Homework Statement I am getting fooled by the this improper integral \int_0^{\infty}\frac{cos(x)+sin(x)}{1+v^2}dv = \pi \cdot e^{-x} How the devil do I go about getting that result? The Attempt at a Solution I end up getting the sum of the two integrals...

I have been bashing my head against this problem for a couple of hours now and cannot for the life of me figure it out. i am able to get AN answer but when i check it with my calculator i always get pi Homework Statement integral from (0, 1) of: (4r*dr)/sqrt(1 - r^4) Homework...

Homework Statement determine the value of the improper integral when using the integral test to show that \sum k / e^k/5 is convergent. answers are given as a) 50/e b) -1 / 5e^1/5 c) 5 d) 5e e)1/50e The Attempt at a Solution f(x) = xe^-x/5 is continuos and positive for all...

Homework Statement Determine how large the number a has to be so that: \int_{a}^{\infty} \frac{1}{1 + x^{2}} dx < 0.001 Homework Equations None. The Attempt at a Solution I tried to evaluate the left hand side and got a final answer of: a > \frac{\pi}{2} - \frac{1}{1000}...

Homework Statement Show that \int_{0}^{\infty}x^{2}e^{-x^{2}}dx = \frac{1}{2}\int_{0}^{\infty}e^{-x^{2}}dx. Homework Equations None. The Attempt at a Solution I used substitution: t = x^{2} dx = \frac{dt}{2x} \frac{1}{2}\int_{0}^{\infty}\sqrt{t}e^{-t}dx Then tried...

Homework Statement Hi. Well, the statement exhorts me to determinate the character of some improper integrals, and it begins with this one \displaystyle\int_{-\infty}^{\infty}xe^{-x^2}dx I really don't know how to work this out. I think its not possible to integrate it, cause its not an...

How do you get this: \int_{-\infty}^{\infty}e^{-(ax^2+b/x^2)}dx = \sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab}} I've been trying all the tricks I know, like differentiating under the integral sign and whatnot, but I can't get it. Thanks.

Homework Statement Use the Comparison Theorem to determine whether the integral is convergent or divergent. \int \frac{2+ e^{-x} dx}{x} from 1 to infinity Wolframalpha tells me this integral diverges, now i just need to know what to compare it to. The Attempt at a Solution So far I've...

Homework Statement \int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}}} Homework Equations None The Attempt at a Solution \int^{1}_{0}\frac{dx}{\sqrt{1-x^{2}}} = sin^{-1}x\right|^{1}_{0} sin^{-1}x\right|^{1}_{0} = \frac{\pi}{2} - 0 so the final answer is just pi/2. I have no problem computing the...

This is the problem: \int_{-\infty}^{\infty} {xe^{-x^{2}}dx} I noticed the function was even, so I then did this: 2\int_{0}^{\infty} {xe^{-x^{2}}dx} I attempted to do integration by parts: u=e^{-x^{2}}, du=-2e^{-x^{2}}, dv=x, v=\frac{x^{2}}{2} which still left me with this at the end: (I...