What is Improper integral: Definition and 238 Discussions
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number,
∞
{\displaystyle \infty }
,
−
∞
{\displaystyle \infty }
, or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration.
Specifically, an improper integral is a limit of the form:
{\displaystyle \lim _{c\to b^{}}\int _{a}^{c}f(x)\,dx,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,dx,}
in which one takes a limit in one or the other (or sometimes both) endpoints (Apostol 1967, §10.23).
By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, perhaps with infinity among the limits of integration. When the definite integral exists (in the sense of either the Riemann integral or the more advanced Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value.
Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function or because one of the bounds of integration is infinite.
I'm trying to solve an improper integral, but I'm not familiar with this kind of integral.
##\int_{\infty}^{\infty} (xa^3 e^{x^2} + ab e^{x^2}) dx##
a and b are both constants.
From what I found
##\int_{\infty}^{\infty} d e^{u^2} dx = \sqrt{\pi}##, where d is a constant
and...
Let ##f:[0;1)\to\mathbb{R}## and ##f\in C^1([0;1))## and ##\lim_{x\to1^}f(x)=+\infty## and ##\forall_{x\in[0;1)}\infty<f(x)<+\infty##. Define $$A:=\int_0^1f(x)\, dx\,.$$ Assuming ##A## exists and is finite, is it possible that ##\text{sgn}(A)=1##?
I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14)
Let ##V## be the set of all real functions ##f##...
Part a: Using the above equation. I got
$$\psi(x) = \int_{\infty}^{\infty} \frac{Ne^{ikx}}{k^2 + \alpha^2}dk $$
So basically I needed to solve above integral to get the wave function. To solve it, I used Jordan's Lemma & Cauchy Residue Theorem.
And obtained $$\psi(x) = \frac {N \pi...
Hi,
I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so.
Integral 1
Now the 1st integral is again integrated
Now the text accompanying the integration...
find out for what values of p > 0 this integral is convergent
##\displaystyle{\int_0^\infty x^{p1}e^{x}\,dx}\;##
so i broke them up to 2 integrals one from 0 to 1 and the other from 1 to ∞ and use the limit convergence test. but i found out that there are no vaules of p that makes both of...
In Physics/Electrostatics textbook, I am in a situation where we have to find the electric field at a point inside the volume charge distribution. In Cartesian coordinates, we can't do it the usual way because of the integrand singularity. So we use the three dimensional improper integral...
Electric potential at a point inside the charge distribution is:
##\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'\delta}
\dfrac{\rho (\mathbf{r'})}{\mathbf{r}\mathbf{r'}} dV'##
where:
##\delta## is a small volume around point ##\mathbf{r}=\mathbf{r'}##
##\mathbf{r}##...
Hi colleagues
This is a very very simple question
I can show when $f$ is integrable and is even i.e. $f(x)=f(x)$ then
$\int_{a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$
what about improper integrals of even functions, like the function ${x}^{2}\ln\left x...
The potential of a dipole distribution at a point ##P## is:
##\psi=k \int_{V'}
\dfrac{\vec{\nabla'}.\vec{M'}}{r}dV'
+k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'##
If ##P\in V'##, the integrand is discontinuous (infinite) at the point ##r=0##. So we need to use improper integrals by removing...
I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it.
I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.
Hi! I am trying to solve problems from previous exams to prepare for my own. In this problem I am supposed to find the improper integral by substituting one of the "elements", but I don't understand how to get from one given step to the next.
Homework Statement
Solve the integral
by...
Hello everyone,
I am stuck on this homework problem. I got up to (ln (b / (b+1)  ln 1 / (1+1) ) but I'm not sure how to go to the red boxed step where they have (1  1 / (b+1) )
if anyone can figure it out Id really appreciate it.
thank you very much.
I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A > R be a continuous function. If f is nonnegative on A, we define the (extended) integral of f over A, as the supremum of all the...
Homework Statement
Determine if the improper integral is divergent or convergent .
Homework Equations

The Attempt at a Solution
When i solved the first term using online calculator , the answer was "The integral is divergent" . However , I got 0 .
Where is my mistake ?
Homework Statement
I have to prove that the improper integral ∫ ln(x)/(1x) dx on the interval [0,1] is convergent.
Homework Equations
I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1.
The Attempt at a Solution
The function can be approximated to ln(x) when it approaches...
15.3.65 Improper integral arise in polar coordinates
$\textsf{Improper integral arise in polar coordinates when the radial coordinate r becomes arbitrarily large.}$
$\textsf{Under certain conditions, these integrals are treated in the usual way shown below.}$
\begin{align*}\displaystyle...
Homework Statement
Let ##f: (1, \infty) \to [0,\infty)## be a function such that the improper integral ##\int_{1}^{\infty} f(x)dx## converges. If ##f## is monotonically decreasing, then ##\lim_{x \to \infty} f(x)## exists.
Homework EquationsThe Attempt at a Solution
This problem doesn't come...
Homework Statement
I have a question.
I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent.
x^2y^2z^2/r^(17/2) * f(x,y,z)dV.
Homework Equations...
Homework Statement
https://holland.pk/uptow/i4/7d4e50778928226bfdc0e51fb64facfb.jpg
Homework Equations
improper integral
The Attempt at a Solution
(attached)
Whats wrong with my calculation?
I cannot figure it out after hours...
Thank you very much!
Homework Statement
Compute the Integral: ##\int_{\infty}^\infty \space \frac{e^{2ix}}{x^2+4}dx##
Homework Equations
##\int_C \space f(z) = 2\pi i \sum \space res \space f(z)##
The Attempt at a Solution
At first I tried doing this using a bounded integral but couldn't seem to get the right...
Homework Statement
Evaluate the indefinite integral as a power series. What is the radius of convergence (R)?
##\int x^2ln(1+x) \, dx##
Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (1)^n \frac {x^{n+3}} {n(n+3)}; R = 1##
Homework Equations
Geometric series
##\frac {1} {1x} =...
Say we have the following result: ##\displaystyle \int_0^{\infty} \frac{\log (x)}{1  bx + x^2} = 0##. We see that the denominator is 0 for some positive real number when ##b \ge 2##. Thus, we obtain a two singularities under that condition. Here's my question. Can we go ahead and say that the...
Homework Statement
Evaluate ##\displaystyle \int_{1}^{\infty} \frac{dx}{(x+a)\sqrt{x1}}##
Homework EquationsThe Attempt at a Solution
First I make the substitution ##u = \sqrt{x1}##, which ends up giving me ##\displaystyle \int_{0}^{\infty} \frac{2u}{u(u^2 + 1 + a)}du##. Here is where I am...
I find it easy to picture bounder integrals because they are the area under the graph but when it is unbounded what is it exactly. If we evaluate it we find the equation of a many possible graphs where the derivative is the function that was originally in the integral. How did we get from that...
206.8.8.11
$\text{determine if the improper integral is convergent} \\
\text{and calculate its value if it is convergent. } $
$$\displaystyle
\int_{0}^{\infty}8e^{8x} \,dx
=e^{8x}+C$$
$$\text{not sure about the coverage thing from the text } $$
My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density ##n## (i.e. there are ##n## loops per length unit), a current of intensity ##I## flows, has the same direction as the loops' moment of magnetic dipole and magnitude ##\mu_0...
Let ##\boldsymbol{l}:\mathbb{R}\to\mathbb{R}^3## be the piecewise smooth parametrization of an infinitely long curve ##\gamma\subset\mathbb{R}^3##. Let us define $$\boldsymbol{B}(\boldsymbol{x})=\frac{\mu_0...
I'm having a tough time with this integral:
$$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$
where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck.
What contour should I...
I've been trying to solve this improper integral ∫[∞][1] ln(x) x^1 dx. I couldn't find any way to use the comparison test to find divergence, so I used substitution and got ∞∞ which I was pretty sure was divergence until I noticed I put 0 instead of 1 making my answer ∞. Do I need to prove...
I think it was Gauss who calculated a limit in two different ways, getting 1/2 one way and infinity the other. As he didn't see the error, he wrote sarcastically, "1/2 = infinity. Great is the glory of God" (In Latin). Anyway, it appears that Wolfram Alpha could do the same thing, as I asked...
Homework Statement
use the comparison theorem to determine whether ∫ 0→1 (e^x/√x) dx converges.
Homework Equations
I used ∫ 0 → 1 (1/√x) dx to compare with the integral above
The Attempt at a Solution
i found that ∫ 0 → 1 (1/√x) dx = 2 ( by substituting 0 for t and take the limit of the...
Homework Statement
∫(sin(x)+2)/x^2 from 2 to infinity. Determine if this improper integral converge or diverge.2. The attempt at a solution
lim(x→infinity)=∫(sin(x)+2)/x^2 from 2 to t.
I know that if the integral ends up to be an infinite number, this will be converge otherwise, it will be...
Homework Statement
Evaluating the following formula: The Attempt at a Solution
Since the integral part is unknown, dividing the case into two: converging and diverging
If converging: the overall value will always be 0
If diverging: ...?
Homework Statement
[/B]
This is the improper integral of which I have to study the convergence.
∫[1,+∞] sinx/x2 dx
The Attempt at a Solution
[/B]
I have tried to use the absolute convergence.
∫f(x)dx converges ⇔ ∫f(x)dx converges
but after i have observed that x^2 is always positive...
I am attempting to solve the improper integral (x*cos^2(x))/(1+x^3) dx between infinity and 1 to see if it converges or diverges. My approach was to place a point 'x' that approaches infinity to be able to solve the integral and then evaluate the limits however i am stuck on actually computing...
Homework Statement
Evaluate the Improper integral [4,13] 1/(x5)1/3
Homework Equations
N/A
The Attempt at a Solution
In step 1, I split the integral into two separate integrals because at x=5, it would be undefined. I made the first limit approach 5 from the left and the second limit...
integral from 2 to infinity dx/(x^2+2x3)
I got this as the result:
lim x to infinity (1/4)(lnx1lnx+1+ln5)
Then I got (1/4)(infinity  infinity + ln5) so do I need to use l'hopital's rule for lnx1lnx+1 or would the final answer be ln5/4? If not, I am unsure of how to...
Homework Statement
Generalize the integral from 0 to 1 of 1/(x^p)
What conditions are necessary on P to make the improper integral converge and not diverge?
I believe I have the answer but I would like to make it more formal and sound. Can someone help me with that?Homework Equations
None...
Homework Statement
http://puu.sh/fYQQj/12819720c6.png
My question is in the attempt at the solution (Number 3)
2. Homework Equations The Attempt at a Solution
I know how to get to lim t→∞ 1/(1p) * (t^(1p)  1^(1p)), I'm not sure what to do to get the 1 instead of 1^(1p) in the above image
The question asks whether the following converges or diverges.
\int_{0}^{\infty } \frac{\left  sinx \right }{x^2} dx
Now I think there might be a trick with the domain of sine function but I couldn't make up my mind on this.
I tried to compare it with 1/x^2, (sinx)/x, and sinx. I actually...
\int_{0}^{\infty} \frac{x^2 dx}{x^5+1}
The question asks whether this function diverges or converges.
I have tried to do some comparisons with x^2/(x^6+1), and x^2/(x^3+1) but it didn't end up with something good.
Then I decided to compare it with \frac{x^2}{x^4+1}
Since this function...
Consider the integral:
$$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$
$R$ is the big radius, $\delta$ is the small radius.
Actually, let's consider $u$ the small radius. Let $\delta = u$
Ultimately the goal is to let $u \to 0$
We can parametrize,
$$z =...
Homework Statement
Improper Integral of theta/cos^2 theta
Homework EquationsThe Attempt at a Solution
Hi all, this was one of the few questions on my final today that I just didn't know how to do. I know how to do trig sub, know all my trig identities and know improper integration, but was a...
State whether the integral converges or diverges and if it converges state the value it converges to.
Integral from 0 to 2 of 1/(1x)dx
I broke it up into 2 integrals (0 to 1) and (1 to 2) set up the limit for both using variables instead of 1 and I evaluated the integral to equal 0 so I...
Homework Statement
Here is a more interesting problem to consider. We want to evaluate the improper integral
\intop_{0}^{\infty}\frac{\tan^{1}(6x)\tan^{1}(2x)}{x}dx
Do it by rewriting the numerator of the integrand as \intop_{f(x)}^{g(x)}h(y)dy for appropriate f, g, h and then reversing...
Homework Statement :
Evaluate: ∫214 (1+X)1/4[/B]
Homework Equations
∫ab f(x)dx =
lim ∫at f(x)dx
t→b
And
∫ab f(x)dx =
lim ∫tb f(x)dx
t→a+
The Attempt at a Solution
So far what I have done is:
(2,1)∪(1,14)
Thus I...
Hello, Two questions will be posed here.
(1) Question about Convergence; quick way.
Hello, I am trying to learn this concept on my own. My major question here is that,
Is there a quick way, to tell if an integral converges or diverges?
Suppose $\int_{0}^{\infty} \frac{x^3}{(x^2 +...