Improper integral Definition and 236 Threads

For $$a\, ,b\in\mathbb{R}\,$$ and $$b>|a|\,$$ show that: $$\int_0^{\infty}\frac{\sinh ax}{\sinh bx}\, dx = \frac{\pi}{2b}\tan\frac{\pi a}{2b}$$

Hi there, I am stuck on this problem: the integral of 1/x^(1/3) from -1 to 8. I have broken it up into the integral from -1 to 0 and 0 to 8. I am confused as to how the negative values under a cubed root affect things and whether or not I need to break it up further. I am not sure whether...

Hey, its me again, just needing someone to verify my findings. Thanks in advance. $$ \int ^{\infty}_0 \frac{1}{e^{3x}} \, dx = \lim_{a\to\infty} \frac{1}{3} \int ^{3a}_0 e^{-u} \, dx$$ u = 3x ,,,,,, du/3 = dx skipping a few steps... $$\lim_{a\to\infty} -\frac{1}{3}e^{-u} |^{3a}_0 = 0 +...

Evaluate the Integral. Just wondering if someone could check my work, thanks in advance. $$\int ^0_{-\infty} \frac{1}{e^{2x}} \, dx $$ $$lim_{a\to-\infty} \int ^0_a \frac{1}{e^{2x}} \, dx = lim_{a\to-\infty} \frac{1}{2} \int ^0_a \frac{1}{e^u}$$ *Letting $$u = 2x$$ && $$du/2 = dx$$ $$...

Need someone to check my work, as well as answer a few questions I'm confused about as well. $$\int ^{\infty}_1 \frac{2x}{(x^2 + 1)^3} \, dx$$ so: $$lim_{a\to\infty} \int^a_1 \frac{2x}{(x^2 + 1)^3} \, dx$$ Letting $$u = x^2 + 1$$ and $$du = 2x \, dx$$ after updating the limits I come up...

Evaluate the Integral. $$ \int^1_{-1} \frac{1}{\sqrt{|x|}} \, dx$$ I know that there is a discontinuity at 0 When they change the limits how are they getting $$\int^0_{-1} \frac{1}{\sqrt{-x}} \, dx + \int ^1_0 \frac{1}{\sqrt{x}} $$ I understand the limit changing part, but I don't understand...

Hey, I need a little help. I'm a bit stuck. Evaluate the Integral $$\int ^{\infty}_2 \frac{1}{x^{1.5}}$$ Here is what I have: $$\int ^{\infty}_2 = x^{-1.5} = \frac{1}{.5} x^{.5} |^\infty _2$$ Am i doing this correctly or no?

Let $$λ \in R$$ $$I=\int_{0}^{\infty} \left(\frac{x+1}{3x^2 + \lambda} - \frac{\lambda}{2x+1}\right)dx $$ I need to find λ for which this would return a number (not infinity) . I tried writing Numerators as derivatives but not sure about the correctness and results.eg...

Homework Statement The problem is attached in this post. Homework Equations The problem is attached in this post. The Attempt at a Solution Lim t -> ∞ ∫ dx/xlnx from 1 to t u-substitution: u=lnx du=1/x dx Lim t -> ∞ ∫ 1/u du Lim t -> ∞ ln u Lim t -> ∞ ln(lnx) from 1 to t Lim t -> ∞...

Hello everyone I need to ask a tough question I have an integral I should plot in 3d from minus infinite to plus infinite it is known that this integral is hard but can be solved numerically I need it in MATLAB is it possible? Thank you

Homework Statement It just wants me to tell whether this is improper or not. [0,infinity] e^(-x^3) dx Homework Equations I say Yes The Attempt at a Solution

I'm supposed to find the integral of f(x) = (e^5x) / (1+(e^10x)) from negative infinity to 0. I know how to set up the integral as the limit as t approaches -∞ of ∫f(x) from t to 0, but I'm stuck on how to actually solve the integral. I've tried by parts and u-sub but I just can't seem to get...

How would I evaluate $$\int_0^\infty \frac{\ln(x)}{1+x^2} dx$$?

Homework Statement 1.Determine the divergence/convergence of the following improper integrals by the evaluation of the limit: \int_{0}^{∞} \frac{dx}{e^{-x} + e^{x}} Homework Equations The Attempt at a Solution Let u = e^x ∴ du = e^x dx I ended up with...

Homework Statement Verify that \int_{ℝ^n}exp(-\frac{λ}{2} \langle Ax, x \rangle-i \langle x,ζ \rangle )dx=(\frac{2\pi}{λ})^{\frac{1}{2}}(detA)^{-\frac{1}{2}}exp(-\frac{1}{2λ} \langle A^{-1}ζ, ζ \rangle ) where A is a symmetric matrix of complex numbers and <ReA x, x> is positive definite, and λ...

At exam today I was to calculate an improper integral of a function f defined by a power series. The power series had radius of convergence r=1. Inside this radius you could of course integrate each term, i.e. symbologically: ∫Ʃ = Ʃ∫ The only problem is that the improper integral went from 0...

Here is a link to the question: Integrate exp(-b(x-a)^2) with respect to x from -infinity to +infinity? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement \int_{0}^{\infty}\frac{\log(x^{2}y^{2}+1)}{y^{2}+1}dyHomework Equations The answer is \pi\log(x+1).The Attempt at a Solution I have attempted many different substitutions like y=\tan\theta. I have also tried breaking up the log but nothing definitive comes out. Any help would...

in this problem I am trying to find the mean of the probability density function of c*e^(-c*t) and by doing so i am multiplying the function stated previously by the variable t, which i know is correct. after taking the anti derivative and evaluating using the limit. I get an indeterminate form...

∞ ∫ x/(x^2+1) dx -∞ I basicaly evaluated the integral and Ln (x^2+1) as the antiderivative and when taking the limits I get ∞-∞ (ln |1| -ln|b+1|) + (ln|n+1|- ln|1|) lim b-> neg. infinity lim n-> infinity does this function converge or diverge? this was a question on...

Homework Statement evaluate the integral 1/(u^2 -36) from 0 to 6 does the integral converge? Homework Equations The Attempt at a Solution integral 1/(u^2 -36) integral 1/((u-6)(u+6)) Partial fraction decomposition 1/((u-6)(u+6)) = A/(u-6) + B/(u+6) 1=A(u+6) + B(u-6) 1=(A+B)u +(6A-6B) A+B=0...

1. integrate from (1 to 3) of function (2) / (x-2)^(8/3) Can someone explain why this diverges. i do not understand it. when i plugged in the numbers there are no discontinuities and this is where i am stuck at. If there are no discontinuity does that means that it diverges? Homework...

Hi guys just want to check my answer for the following improper integral. ∫(2 to ∞) dv/v^2+2v-3. After doing partial fractions, integrating and evaluating I got the following for the answer: 0-(1/4)ln(1/5) How does this compare to other answers? Is there a way I can accurately...

Hi guys just want to check my answer for the following improper integral. ∫(2 to ∞) dv/v^2+2v-3. After doing partial fractions, integrating and evaluating I got the following for the answer: 0-(1/4)ln(1/5)=(1/4)ln(1/5) How does this compare to other answers? Is there a way I can accurately...

Here is the question: Here is a link to the question: Improper integral help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

How would one go about computing the following improper integral, with limits of integration [0,∞) using residues? \int exp(x+1/x)/x

Homework Statement \displaystyle \int_{1}^{\infty} 1/(x^2+ 3 \ |sin x| +2) dx Homework Equations N/AThe Attempt at a Solution \displaystyle \int_{1}^{\infty} 1/(x^2+ 3 \ |sin x| +2) dx = \displaystyle lim_{t\rightarrow \infty} \int_{1}^{t} 1/(x^2+ 3 \ |sin x| +2) dx Side Work...

Homework Statement \int_{-\infty}^{0} 2^{r}dr Homework Equations The Attempt at a Solution \int_{-\infty}^{0} 2^{r}dr = \lim_{t \to -\infty} \int_t^0 2^{r}dr=\lim_{t \to -\infty} \frac{2^{r}}{ln2}|_{t}^{0} = \lim_{t \to -\infty} \frac{1}{ln2}-\frac{2^{t}}{ln2} Which I thought = ∞, but I guess...

Sir, Recently when i am evaluating a convolution integral, i came across the integral of |sinx/x| under limits running from 0 to infinity. when i tried to evaluate the integral, i used complex analysis tools like assuming a function e^(iz) / z and deduce the above integral from integral of...

\begin{array}{l} \int\limits_{ - \infty }^\infty {\frac{{{x^2}}}{{{x^6} + 9}}} \\ = \int\limits_{ - \infty }^0 {\frac{{{x^2}}}{{{x^6} + 9}}} + \int\limits_0^\infty {\frac{{{x^2}}}{{{x^6} + 9}}} \\ = \mathop {\lim }\limits_{t \to - \infty } \int\limits_t^0 {\frac{{{x^2}}}{{{x^6} + 9}}}...

Homework Statement Prove that ##\int_0^{\infty} sin(x^2)dx ## exists The Attempt at a Solution I split the integral in two parts: ##\int_0^1 sin(x^2)dx## which exists because ##sin(x^2)<1## and so ##\int_0^1 sin(x^2)dx < \int_0^1 1= 1*(1-0)## but i don't know how to do with the...

Homework Statement Evaluate the integral: ∫(0 to ∞) [dv/((1+v^2)(1+tan^-1(v))] Homework Equations U-substitution, taking limit to evaluate improper integralsThe Attempt at a Solution http://imgur.com/CjkRF As you can see in the image, I try u-substitution and then take the integral. I end...

Homework Statement Evaluate if the integral diverges or converges using the comparison theorem. \int^ \infty_2 \frac{dx}{\sqrt{x^3+1}} Having trouble with this question, the exercises I have managed I generally guessed if it was convergent/divergent, and then found a smaller of bigger...

→Homework Statement Integrate the improper integral (use correct notation). State whether it's converging or diverging. 10 ∫ 7/(x-9)^2 dx 8 Homework Equations b c ∫ f(x) dx= lim ∫ f(x) dx a c → d a The Attempt at a Solution...

We have that \int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}\int^{1}_{t}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}[arcsin(x)]^{1}_{t}=\frac{\pi}{2} However, I think \int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}} should equal to lim_{\stackrel{}{t...

Homework Statement evaluate $$\int_0^1\frac{Ln(x)}{1+x}\,dx$$ Homework Equations I know the way to solve most improper integrals; replacing 0 or the bound causing the issue with a variable and have the limit of the integral as the variable goes to infinity. My question is using...

Homework Statement ∫-∞∞(dx/x2) Homework Equations The Attempt at a Solution ∫(dx/x2) = -1/x (-1/∞) - (-1/-∞) = 0 However, the answer is that the integral diverges. Why is this the case?

Hi all, This is a case of a book answer going against Wolfram's and my answer. The problem is ∫∞e(ln(x)/x)dx The book claims the answer is ∞. I would think it is a case of ∞/∞ and use L'Hospital's Rule. Wolfram has the same solution. *= lima->∞(1/x)/1 = 0 Which would be correct?

State if the following integral converges or diverges, and justify your claim. \int_{-1}^{1} \frac{e^x}{x+1}\,dx I tried using the comparison theorem by comparing it to \frac{1}{x+1} . But for the interval (-1,0) the function is smaller for all x. So I could not conclude whether it...

∞ ∫xe^[-x^2] dx -∞ So basically I've solved for everything in this problem and it looks like it should be an indeterminate form and thus divergent. My book and Wolfram both say it's 0 and convergent though. I get it down into: lim [[e^(-t^2)] - e^0]/2 + lim [e^0 - [e^(-v^2)]]/2...

Let f(x) be a continuous functions on [0,∞) and that ∫ |f(t)|^2dt is convergent for 0≤t<∞. Let ∫ |f(t)|^2dt for 0≤t<∞ equals F. Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F. I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone...

Homework Statement [SIZE="4"]\int \frac{dx}{\sqrt{x^2-4}} Homework Equations The Attempt at a Solution I tried trig-substitution, by realizing that [SIZE="4"]cot\theta = \frac{4}{\sqrt{x^2-4}} and that [SIZE="4"]-4sin\theta = dx My answer, though, found after the...

\int_0^1 \frac{1}{\sqrt{x}}\,\mathrm{d}x = \lim_{\varepsilon \to 0+}\int_\varepsilon^1 \frac{1}{\sqrt{x}}\,\mathrm{d}x My question is about the usage of 0+ in the limit.(I evaluated the integrals and arrived at the part where I substitute upper and lower limits.) Did the author...

I understand most of the problem, but have yet to understand where a particular term came from. The problem is as follows: Homework Statement Show that (0 to ∞)[SIZE="5"]∫[SIZE="4"]dx/[(x2+1)√x] = π/√2 Hint: f(z)=z−1/2/(z2+ 1) = e(−1/2) log z /(z2+ 1). The Attempt at a Solution I actually...

Homework Statement I'm trying to test whether the sequence converges or not: \sum^{∞}_{k = 1}ke^{-2k^2} 2. The attempt at a solution I tried to evaluate this in two ways, each of which produced different answers. I was able to eventually discover that this series does converge, but I still...

Homework Statement Discuss for alpha the convergence of the following improper integral: \displaystyle \int\limits_{0}^{3}{\frac{{{x}^{3\alpha }}}{{{\left( 9-{{x}^{2}} \right)}^{\alpha }}}} Homework Equations The Attempt at a Solution Well, my attempt was to simplify the integral to...

Thanks to those who participated in last week's POTW! Here's this week's problem. ----- Problem: Show that \[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sqrt{x^2+y^2+z^2}e^{-(x^2+y^2+z^2)}\,dx\,dy\,dz = 2\pi\] (Note that the improper triple integral is defined as...

Homework Statement (a). Prove that the improper integrals converge: \displaystyle \int\limits_{0}^{1}{\frac{\ln x}{1+{{x}^{2}}}}dx \displaystyle \int\limits_{1}^{\infty }{\frac{\ln x}{1+{{x}^{2}}}}dx And relate each other. (b) Deduce the value of: \displaystyle...

So recently I've been working through some challenge problems from my old calculus textbook for fun. I'm stuck on one of the integrals, though, and can't find any solutions online. This isn't for homework...it's for my interest and hopefully the interest of others. Here it is (sorry about the...

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...