# 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:

lim

b

a

b

f
(
x
)

d
x
,

lim

a

a

b

f
(
x
)

d
x
,

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

lim

c

b

a

c

f
(
x
)

d
x
,

lim

c

a

+

c

b

f
(
x
)

d
x
,

{\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. ### 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. ### 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. ### 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##...

31. ### 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. ### 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. ### 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. ### Comparison Test for improper integral

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

Homework Statement Evaluate the Improper integral [4,13] 1/(x-5)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...
40. ### MHB Improper integral and L'Hôpital's rule

integral from 2 to infinity dx/(x^2+2x-3) I got this as the result: lim x to infinity (1/4)(ln|x-1|-ln|x+1|+ln|5|) Then I got (1/4)(infinity - infinity + ln|5|) so do I need to use l'hopital's rule for ln|x-1|-ln|x+1| or would the final answer be ln|5|/4? If not, I am unsure of how to...
41. ### Generalize improper integral help

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...
42. ### Confused at a fairly simple step in an improper integral

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/(1-p) * (t^(1-p) - 1^(1-p)), I'm not sure what to do to get the 1 instead of 1^(1-p) in the above image
43. ### Improper integral comparison test

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...
44. ### Improper integral comparison test

\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...
45. ### Proving integral on small contour is equal to 0.

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 =...
46. ### Improper Integral of theta/cos^2 theta

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...
47. M

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

State whether the integral converges or diverges and if it converges state the value it converges to. Integral from 0 to 2 of 1/(1-x)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...
48. ### Evaluating an Improper Integral using a Double Integral

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...
49. ### Evaluate improper integral: Discontinuous integrand

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...
50. ### MHB Improper Integral Question (convergence & evaluation)

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 +...