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 _{b\to \infty }\int _{a}^{b}f(x)\,dx,\qquad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx,}
















{\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.

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  1. H

    Improper integral of a normal function

    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...
  2. Rlwe

    I Is the sign of the integral of this function negative?

    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##?
  3. H

    Prove that the inner product converges

    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##...
  4. X

    Normalizing wavefunction obtained from Lorentzian wave packet

    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...
  5. O

    MHB Can Improper Integrals Help Solve This Inequality?

    This is my method, could you help me to continue?
  6. C

    A Evaluation of an improper integral leading to a delta function

    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...
  7. Yohan

    Finding if an improper integral is Convergent

    find out for what values of p > 0 this integral is convergent ##\displaystyle{\int_0^\infty x^{p-1}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...
  8. Beelzedad

    I Is my interpretation of this three dimensional improper integral correct?

    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...
  9. Beelzedad

    I Is Leibniz integral rule allowed in this potential 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}##...
  10. S

    MHB Improper integral of an even function

    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...
  11. M

    I Why ignoring the contribution from point r=0 in eq (1) and (2)?

    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...
  12. Zack K

    I Why can an infinite area have a finite volume or SA?

    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.
  13. ChristinaMaria

    Improper integral with substitution

    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...
  14. Y

    MHB Improper integral from 1 to infinity

    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.
  15. Bill2500

    I Munkres-Analysis on Manifolds: Extended Integrals

    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 non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...
  16. F

    Determine if the improper integral is divergent or not

    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 ?
  17. Cathr

    Improper integral convergence from 0 to 1

    Homework Statement I have to prove that the improper integral ∫ ln(x)/(1-x) 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...
  18. karush

    MHB Evaluating Improper Integrals in Polar Coordinates

    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...
  19. B

    Improper Integral of a Monotonic Function

    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...
  20. C

    Improper integral with spherical coordinates

    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...
  21. yecko

    Solve Improper Integral Homework - Get Help Now!

    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!
  22. T

    Another Improper Integral Using Complex Analysis

    Homework Statement $$\int_{-\infty}^\infty \space \frac{cos(2x)}{x-3i}dx$$ Homework EquationsThe Attempt at a Solution $$\int_{-R}^R \space \frac{e^{2ix}}{x-3i}dx + \int_{C_R} \space \frac{e^{2iz}}{z-3i}dz = 2\pi i\sum\space res \space f(z)$$ Then using Jordan's Lemma, as ##R\to\infty## the...
  23. T

    Improper Integral Using Complex Analysis

    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...
  24. uchuu-man chi

    Need help evaluating an improper integral as a power series.

    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} {1-x} =...
  25. Mr Davis 97

    I Telling whether an improper integral converges

    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...
  26. Mr Davis 97

    Integral of $\frac{1}{(x+a)\sqrt{x-1}}$ - Solution

    Homework Statement Evaluate ##\displaystyle \int_{1}^{\infty} \frac{dx}{(x+a)\sqrt{x-1}}## Homework EquationsThe Attempt at a Solution First I make the substitution ##u = \sqrt{x-1}##, which ends up giving me ##\displaystyle \int_{0}^{\infty} \frac{2u}{u(u^2 + 1 + a)}du##. Here is where I am...
  27. NihalRi

    I What is the improper integral?

    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...
  28. karush

    MHB Is the Improper Integral Convergent and What is its Value? $\text{determine if the improper integral is convergent} \\ \text{and calculate its value if it is convergent. } $ $$\displaystyle \int_{0}^{\infty}8e^{-8x} \,dx =-e^{-8x}+C$$ $$\text{not sure about the coverage thing from the text } $$
  29. DavideGenoa

    I Properties of ideal solenoid: postulates or derivations?

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  30. DavideGenoa

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  31. J

    I Need help with this definite integral

    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...
  32. S

    I Unsure of solution to improper integral

    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...
  33. nomadreid

    Improper integral done two different ways

    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...
  34. S

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  35. P

    Improper Integral: ∫(sin(x)+2)/x^2 from 2 to ∞ - Converge or Diverge?

    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...
  36. J

    What Happens When Evaluating Improper Integrals with Limit?

    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: ...?
  37. mekise

    Improper Integral Sinx/x^2 and similar with sinx

    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...
  38. B

    MHB Improper integral convergence/divergence

    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...
  39. T

    Evaluate the Improper integral [4,13] 1/(x-5)^(1/3)

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  40. rayne1

    MHB Improper integral and L'Hôpital's rule

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  41. T

    Generalize improper integral help

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  42. G

    Confused at a fairly simple step in an improper integral

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  43. H

    Improper integral comparison test

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  44. H

    Improper integral comparison test

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  45. A

    Proving integral on small contour is equal to 0.

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  46. L

    Improper Integral of theta/cos^2 theta

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  47. M

    Improper Integral: Convergence and Divergence Analysis for 1/(1-x) from 0 to 2

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  48. W

    Evaluating an Improper Integral using a Double Integral

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  49. H

    Evaluate improper integral: Discontinuous integrand

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  50. A

    MHB Improper Integral Question (convergence & evaluation)

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