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.
Homework Statement
f(x) is a continuous and positive function when x\in[0,\infty). (#1)
x_n is a monotonic increasing sequence, x_0=0 ,x_n \rightarrow \infty. (#2)
Prove or contradict:
\mbox{If } \sum_{n=0}^\infty \int_{x_n}^{x_(n+1)} f(x)dx \mbox{ is convergent (#3) then }...
Homework Statement
evaluate
\int \frac{1}{(1+x)\sqrt{x}} dx
Homework Equations
N/A
The Attempt at a Solution
how to do this, i can't use partial fraction, and integration by part makes it harder i guess, maybe using substitution, but how T_T
helpp
Homework Statement
Let [a, b) be an interval in the reals, with -\infty < a < b \leq \infty , and let \alpha: [a,b) \to \mathbb{R} be monotone increasing. Suppose that f: [a,b) \to \mathbb{R} is a function such that for each c \in (a,b) , f is integrable over [a,c ] with respect to...
Homework Statement
Solve the integral \int\frac{1}{\sqrt[3]{x-1}}. Upper limit of integration is 1 while lower limit is 0.
Homework Equations
N/A.
The Attempt at a Solution
The only thing that I'm sure about is that the antiderivative of the integral is \frac{3}{2}(x-1)^(2/3) +...
Homework Statement
integral of sech(x) from -Inf to Inf using residues.
Homework Equations
Calculate using: (2 Pi I) * Res[sech(x), "poles in upper half plane"]
The Attempt at a Solution
I used sech(x) = 2/[exp(x)+exp(-x)] to find a simple pole at z = (I Pi)/2 with a residue of -I. Then...
Hello--
I have a function:
u(t,\tau)=\frac{1}{\pi}\int_{0}^{\infty}\! G(\omega)\, d\omega
G(\omega)=4\sqrt{\pi}\frac{\omega^{2}}{\omega_{0}^{3}}\mbox{exp}\left(-\frac{\omega^{2}}{\omega_{0}^{2}}\right)\mbox{cos\left(\omega...
Homework Statement
Homework Equations
The Attempt at a Solution
I'm trying to rewrite the integral as shown
Most probably a real simple answer
Thank you
Hello--
I need to generate synthetic data to test an algorithm used to process data from an experiment. A synthetic wavelet is constructed using the following equations, but I am uncertain how to numerically evaluate the improper integral shown below.
\[
u(t) = {\mathop{\rm...
Homework Statement
Using the fact that the integral from -Infinity to Infinity of e^-x^2 is equal to Sqrt(Pi), find the integral from -Infinity to Infinity of x^2 * e^-x^2
Homework Equations
The Attempt at a Solution
I really don't know how to find this using the fact that...
Homework Statement
By rotating R=$\{ (x,y)|x\geq0, 0\leq y\leq \frac{1}{0.6 x+1.7}\}$ about the x-axis we obtain a solid with the volume V = ______
Homework Equations
The Attempt at a Solution
$\int _0^{\infty }\frac{dx}{0.6 x+1.7}$ is divergent
but what do i do to get the volume? if don't...
Homework Statement
I am to determine whether the following integral is convergent or divergent
\int_0^1 \frac{sin(x)}{x}
From what I hear since, lower limit is zero there is a removable discontinuity.
Thus just because of this, it is convergent? Can someone let me know if this
is correct.
Homework Statement
\int(2dx/(x^2+4)
from x= -\infty to x=2
Homework Equations
No specific ones.
The Attempt at a Solution
So, from there I tried to split the integral into two, integrating between 2 and -2, and -2 and -\infty, but I got very lost trying to take the limits for these, partly...
How the following limit can be zero, since after applying L'Hospital rule the root x will be in the numerator, which together with sinx will constitute infinity.
Homework Statement
Use Comparison Theorem to determine whether the integral is convergent or divergent:
integral from 0 to infinity of: arctan(x) / (2 + e^x)
Should look like this: http://bit.ly/cAhytV
Homework Equations
--
The Attempt at a Solution
I tried to compare...
[Solved] Improper Integral Integration
Sorry, don't know how to use the latex stuff for integrals :P
Homework Statement
Integrate the following from 0 to infinity: 1/(sqrt[x]*(1+x))
Homework Equations
Integrate 0 to 1: 1/(sqrt[x]*(1+x))
Integrate from 1 to infinity...
Homework Statement
Use a comparison to determine if the improper integral converges or diverges. If the integral converges, give an upper bound for the value.
Integral of d(theta) / (theta^3 + theta)^1/2 from 1 to infinity
Homework Equations
N/A
The Attempt at a Solution
I'm...
Homework Statement
\int_{-\infty}^{\infty} e^{x\over 2}e^{-x^2\over 2} dx
Homework Equations
\int_{-\infty}^{\infty}e^{-x^2\over a} dx = \sqrt{\pi\over a} a>0
The Attempt at a Solution
Can't seem to penetrate it, I thought about trying to isolate the second term with...
Homework Statement
\int_{-3}^{1}\frac{x}{\sqrt{9-x^{2}}}
Homework Equations
Let f be continuous on the half-open interval (a, b] and suppose that
\lim_{x \to a^{+}} |f(x)| = \infty. Then
\int_{a}^{b}f(x) dx = \lim_{ t \to a^{+}}\int_{t}^{b}f(x)
dx
The Attempt at a Solution...
if you use the limit comp test to show a polynomial behaves like it's highest powered term and that term has a negitive coefficient can the test still be used but with abs vals
\int_{-1}^1 \frac{dx}{x} = \lim_{a \rightarrow 0} \int_{-1}^a \frac{dx}{x} + \int_a^1 \frac{dx}{x} = 0 .
Since 2a also goes to 0 for 'a' going to 0, then we have \int_{-1}^1 \frac{dx}{x} = \lim_{a \rightarrow 0} \int_{-1}^{2a} \frac{dx}{x} + \int_a^1 \frac{dx}{x} = ln2.
It seems like...
Homework Statement
Find:
\int_{0}^\infty \frac{\sqrt[3]{x}-\sqrt{x}}{x^b+a^b}dx
With x>0 and a>2.
Homework Equations
The Attempt at a Solution
I think it's probably something with Beta, but I'm not sure how to change it into the proper form.
I thought changing it into:
\int_{0}^\infty...
Hey guys, I was doing some homework problems and I ran into a problem regarding how to solve a certain improper integral.
\int e^{t*(b-s)} evaluated from 0 to \infty
So I take the integral and get
\frac{\int e^{t*(b-s)}}{-(b-s)} which evaluated from 0 to \infty
gives me 0 -...
Establish convergence/divergence of the following improper integral:
integral from 0 to infinity of 1 / ( x^(1/3)*(/x-5/^(1/3))*(1 + sqrt(x))^0.7) )
My attempt at a solution was to break it up into 3 intergrals: 0 to 1, 1 to 5, and 5 to infinity...I showed that the first two of these...
Homework Statement
AN object moving along a curve in the xy-plane is at position (x(t),y(t)) at time t, where
dx/dt=Arcsin(1-2*e^(-t)) and dy/dt= 4t/(1+t^3)
for t>or= 0. At time t=2, the object is at the point (6,-3).
a. Let m(t) denote the slope of the line tangent to the curve at...
Homework Statement
The integral from -infinity to infinity of (2-x^4)dv
Homework Equations
U substitution
The Attempt at a Solution
Dont know what to use as my "u" ?
Can someone please help me out? Thank you in advance.
given the improper integral from 0 to 1 of
\intdx/\sqrt[3]{x}(ee-e-x)
i am asked if it comverges or diverges,
what i have learned is that if i can find :
-a similar function that is bigger than my function and that i know converges, then my function also converges
-a similar function...
Homework Statement
integral from 2 to infinty 1/(x-sqrt(x))
The Attempt at a Solution
my teacher wants us to compare it to another function in the form 1/x^p
and not integrate it so
would i compare it to 1/x and then do the limit comparison test
limit as x approaches infinity...
Homework Statement
Prove that \lim_{x \rightarrow \infty} exp(-x^2) \int_0^x exp(t^2) dt = 0 .
Homework Equations
The Attempt at a Solution
This question is giving me a lot of difficulty. I've tried a lot of different ways to do it, here is a list of ways that I've tried...
Homework Statement
Prove that \int_0^{\infty} sin^2(\pi(x + 1/x))dx does not exist.Homework Equations
The Attempt at a SolutionFirst, we can construct a sequence as follows:
\int_0^{\infty} f(x)dx = \lim_{n \rightarrow \infty} S_n , where S_n =\int_0^{1} f(x)dx, \int_0^{2} f(x)dx...
Homework Statement
Integral from negative infinity to positive infinity of (1/(sqrt(1+x^2))dx
2. The attempt at a solution
Using trig substitution I got the integral equal to ln|sqrt(1+x^2) + x| Finding this was not the difficult part. Evaluating it is.
I set it up like this: lim b -->...
Homework Statement
I'm trying to show that this improper integral converges
\int_{0}^{1} \sin \left ( x + \frac{1}{x} \right )dxHomework Equations
The Attempt at a Solution
I thought a comparison test would be nice but I can't think of the right one if that is the way to go. I don't think a...
Homework Statement
integral from -1 to 1 of 3/x^2 dx
Homework Equations
The Attempt at a Solution
Limit
t --> infinity integral from -1 to t of 3/x^2 dx + limit t --> infinity integral from t to 1 of 3/x^2 dx
Limit
t-> infinity -3/t - (-3/-1) + Limit t-->...
Homework Statement
Determine if the integral converges or diverges?
it;s the integral of 0 to infinity
of 1/(1+x^6)^(1/2)
Homework Equations
so I compared it with 1/x^2
The Attempt at a Solution
the answer key says it converges but i think it diverges since the integral of 1/x^2...
Hello and Happy New year, I'm having some trouble computing this integral:
limn->00\int^{1}_{0}\sqrt[3]{1+x^{n}sin(nx)}
Any suggestions are appreciated.
Homework Statement
integral of x*e^x from 0 to infinity
Homework Equations
N/a
The Attempt at a Solution
i first integrated the integrand and got (xe^x - e^x) then i use limit as R approaches infinity and used 0 and R as my limits. when calculating i keep getting 1 instead of...
1. Let F(x)= \int^{x}_{0} \frac{sint}{t} and f(x) = \frac{sinx}{x}. If x approaches infinity, F(x) approaches \pi/2. So, Explain what does this mean for the improper integral \int^{\infty}_{o}\frac{sinx}{x}
Homework Equations
Explain what does this mean for the improper integral...
Homework Statement
integral (from 0 to 1) of (lnx)dx/(x^0.5)
Homework Equations
I did u-substitution and got the antiderivative to be 4ln(sqrt(x)) - 4sqrt(x)
The Attempt at a Solution
The answer that I got was that the limit of the antiderivative (as t approaches 0 from the...
Homework Statement
i was deriving an infinite line of charge formula by coloumb's law:
so i got stuck with this integral (since it is in the maths forum)
\vec{E}_{\rho} = \int_{-\infty}^{\infty} \frac{\rho_L \rho dz}{4\pi\epsilon_o ({\rho}^2 + z^2)^{\frac{3}{2}}} Homework Equations
where...
This was a problem on one of my previous tests that I got wrong entirely. In preparing for my final, I'm attempting to redo it. I was wondering if someone could check my work.
Determine whether the following improper integral is convergent or divergent. If the integeral is convergent, find its...
Let A be a constant.
Let f(t) be an integrable function in any interval.
Let h(t) be defined on [0, oo[ such that
h(0) = 0
and for any other "t", h(t) = (1 - cos(At)) / t
It is not difficult to see that h is integrable on [0, b] for any positive "b", so fh is also integrable in...
Homework Statement
Evaluate the integral: \int\frac{dx}{x^{3}+x^{2}+x+1}
from infinity to zero
Homework Equations
lim t--> infinity [/tex] \int \frac{dx}{x^{3}+x^{2}+x+1}
The Attempt at a Solution
lim t-->infinity [/tex] \int \frac{dx}{(x+1)(x^{2}+1}
I'm stuck on where to go...
Could anyone please explain how to solve the improper integral below?
I have no idea of how to do it.
If f is a bounded non-negative function, then show that
the integral from zero to infinity of f(x+1/x)*ln(x)/x dx=0.
Thank you.
Improper Integral [Solved]
Homework Statement
\int_{0}^{\infty} (x-1)e^{-x}dx
Homework Equations
Integration by Parts
Improper Integrals
The Attempt at a Solution
\lim_{R\rightarrow \infty} \int_0^R~xe^{-x}-e^{-x}dx
Let u = x
du = dx
Let dv = e^-x...
Homework Statement
Show \mathop{\lim}\limits_{n \to \infty}(\frac{1}{n!}\int_{1}^{\infty}x^n\frac{1}{e^x} dx )=1
Homework Equations
The hint is that e=\mathop{\lim}\limits_{n \to \infty}\sum_{k=0}^{n}1/k!
The Attempt at a Solution
First I wrote out the improper integral as limit...
Homework Statement
Does the integral from 0 to Infinity of \int\frac{1}{\sqrt x \sqrt{x+1}\sqrt{x+2}}dx converge or diverge?Homework Equations
None.The Attempt at a Solution
I tried to integrate it, but I haven't even been able to do that so I couldn't then evaluate the limit of the...