Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [improper-integrals]

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.

An improper integral is defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.

Specifically, an improper integral is a limit of the form:

$$\lim_{b\to \infty} \int_{a}^{b} f(x) \ dx \,,\ \lim_{a \to -\infty} \int_{a}^{b} f(x) \ dx$$ or of the form $$\lim_{c \to b^{-}} \int_{a}^{c} f(x) \ dx \,,\ \lim_{c \to a^{+}} \int_{c}^{b} f(x) \ dx$$

in which one takes a limit in one or the other (or sometimes both) endpoints.

Often, we can compute values for improper integrals, even when the function cannot be integrated 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.

7820 questions
0
votes
1 answer

Computing Impropoer Integral Of Gamma Distribution

I am trying to compute $$ \begin{align*} \mathrm{E}[X^2] &= \lim_{t\to\infty} \int_{0}^{t} x^2 \frac{\lambda^rx^{r-1}\exp(-\lambda x)}{\Gamma (r)}dx \\[2em] &= \frac{\lambda^r}{\Gamma (r)} \lim_{t\to\infty} \int_{0}^{t} x^{r+1}\exp(-\lambda…
user13317
  • 181
0
votes
2 answers

Improper integrals using comparison theorem

In their working out I understand for the numerator 2+cosx=3 as cosx is less than equal to 1 but in the denominator I don't understand how they got from 3 square root x-x squared sinx all the way upto 2 square root x, I don't understand any of the…
user134785
  • 1,117
0
votes
1 answer

An explanation of the integration

So, the integral is: $$\int_1^2\frac{x-2}{\sqrt{x-1}}dx$$ If I copied correctly from the board, the teacher said if x approaches 1+, the function approaches +$\infty$. What is the difference between 1- and 1+? If x approaches 1+, isn't the function…
A6SE
  • 1,645
0
votes
0 answers

whether $\int_2^{\pi/2}\log\sin x \,dx$ proper or improper

My book suggests $$\int_2^{\pi/2}\log\sin x \,dx$$ is an improper integral. But I think it is not for it is bounded on the respective interval....am I correct?
user210709
  • 43
0
votes
1 answer

Definite integral with Infinite border?

I am instructed to use definite integral to calculate things such as $\int_{a=0}^{b=\infty}x^{7}$ but $\Delta x=\frac{a-b}{n}$ diverges. PatrickJMT shows here to use the border-difference divided by n for $\Delta x$, what should I use now? Can I get…
hhh
  • 5,469
0
votes
2 answers

Integral of 1/sinx between 0 and 1 diverges.

I am learning about ways to test if an integral converges or diverges and I am stuck with this one: $\displaystyle{\int{{\rm d}x \over \sin\left(\, x\right)}}$ between $0$ and $1$. The tests I know are: The Direct Comparison Test. The 2 Way Limit…
user187039
-1
votes
2 answers

How to find the convergence of an improper integral with finite limits?

$$\int_{0}^2\frac{\sin2x}{2x^2-\pi x}dx$$ I simply have to find the convergence of this integral between [0,2]. First I was tasked with doing an integral decomposition, since this integral is undefined at x=0 and x= pi/2. The decomposition that the…
Spencer Ken
  • 1
-1
votes
1 answer

How do I prove that $\int^{+\infty}_0\frac{x}{(1+x^7)^{1/3}}dx$ converges?

How do I prove that $\int^{+\infty}_0\frac{x}{(1+x^7)^{1/3}}dx$ converges? I wanted to do a comparison test with the integral $\int^{+\infty}_0\frac{x}{(1+x)^{1/3}}dx$, but it diverges. I can't seem to find a function which I would compare this to.…
user926287
-1
votes
1 answer

Determine if the following integral is convergent or not

I want to determine the convergence of the following improper integral $$ I := \int_{0}^{+\infty}\frac{\sin(x)}{x^2-x}dx. $$ What I did is to rewrite the integral as $$ \int_{0}^{+\infty}\frac{\sin(x)}{x^2-x}dx = \int_{0}^{1}\frac{\sin(x)}{x^2-x}dx…
user_12345
  • 348
-1
votes
1 answer

Integral over $[-1,0 ]$ with parametrized exponent

The question is when the following integral converges or diverges: $\int_{-1}^{0} x^{-p}$ for $p > 0$. I am stuck at the case where $0 < p < 1$, where I end up with $$\frac{1}{1-p} - \frac{(-1)^{1-p}}{1-p}$$ I am not sure how to split this funciton.…
AlphaDJog
  • 107
-1
votes
2 answers

Is $\lim_{a\to -\infty}\int_{-\infty}^a f(x)\, dx=0$?

Suppose we have an integral like $$ \int_{-\infty}^a f(x)\, dx $$ and want to know if the integral converges as $a\to -\infty$. Is it correct that this is necessarily converging to $0$, that is $$ \lim_{a\to -\infty}\int_{-\infty}^a f(x)\,…
Scuderi
  • 320
-1
votes
2 answers

Using the Euler-Poisson integral, prove the identity

We have: $$I(a) =\int_{-\infty}^{+\infty}e^{-(ax^2+2bx)}dx $$ To prove: $$I(a) = \sqrt{\frac\pi a}e^{b^2/a}$$ I tried to differencate both sides, and got this: Left side: $$I'(a) = -2a\int_{-\infty}^{+\infty}xe^{-(ax^2+2bx)}dx -I(a)$$ Right…
Egor
  • 283
-1
votes
3 answers

Integral of $\int_0^\infty {x^2}{e^{-3x}}\,dx$

$$\int_0^\infty {x^2}{ e^{-3x}}\,dx$$ What I attempted was integration by parts twice, but I end with $-\frac{2}{27}$. That's obviously wrong, should be positive. I am also unsure whether or not $$\lim_{t\to \infty} \frac{t^2}{e^{3t}}$$ converges to…
HashFuncMap
  • 51
-1
votes
2 answers

Evaluating $\int_{1}^\infty\frac{1}{t^a\sqrt{t^2-1}}dt$ for $a\geq 1$

I know that the this integral converges, but I can't show it. And, how can I proceed to calculate its value? $$\displaystyle \int_{1}^\infty \dfrac{1}{t^a\sqrt{t^2-1}} dt\qquad (a \geq 1)$$
Suna
  • 55
  • 7
-1
votes
3 answers

What do improper integrals represent?

So I was wondering what is the "meaning" behind the improper integral, say $\int_0^{\infty}f(x) dx$. Usually, an integral of a function over a certain region is simply the area under the function, however, in this case, it doesn't seem fully the…
Sorfosh
  • 3,266
1 2 3
…
26
27