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 ∞ or −∞, or as both endpoints approach limits.

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2answers
75 views

Integral of $x^2e^{-ax^2}$

Hey guys I need to find the following integral using integration by parts and not the gamma function. Also there is an a constant a in the exponential function. So it is actually $x^2e^{-ax^2}$. ...
3
votes
5answers
204 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
3
votes
2answers
81 views

Proving an inequality on $ \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$

EDIT : I have posted a proof below that needs to be reviewed. Some definitions Let $$\begin{array}{ccccc} f & : & \mathbb R_+^* & \to & \mathbb R_+^* \\ & & x & \mapsto ...
0
votes
1answer
52 views

How can I find this integral

How can I find this integral? please help $$I=\int_0^1\frac{e^{-\sqrt{x}}}{\sqrt{x}}\ dx.$$
1
vote
1answer
52 views

integral with gaussian function

I am trying to evaluate the following integral: $$ \int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z, $$ where the integration is w.r.t. to $z$, and the ...
2
votes
0answers
95 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
2
votes
1answer
60 views

Improper integral $\int^{1}_{0} \frac{x}{\sin{(x^{p})}} dx$

I have an improper integral with $p > 0$, $$\int^{1}_{0} \frac{x}{\sin{(x^{p})}} \ dx$$ and I want to find for which $p$ the integral exists. Now we should consider when $p = 1$ and when $p \not= ...
2
votes
1answer
88 views

Prove that $\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n!\zeta (n+1)$

I have encountered the following identity on Wolfram alpha and I fail proving it (with $n \in \mathbb N^*$) $$\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n! \zeta(n+1)$$ I tried to rewrite the ...
1
vote
1answer
47 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
2
votes
2answers
106 views

Which my step is incorrect to prove $\int_{-\infty}^\infty xe^{-x^2}dx = 0$?

To prove $\int_{-\infty}^\infty xe^{-x^2}dx = 0$, my solution is Let $y=x^2$, then $dy=2xdx$. $\begin{align} \int_{-\infty}^\infty xe^{-x^2}dx &= \frac{1}{2}\int_{-\infty}^\infty e^{-x^2}2xdx ...
11
votes
1answer
251 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$$ It appears to be ...
18
votes
4answers
2k views

A difficult integral

How do I find $$\large\int_{0}^{\infty}e^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive numbers. UPDATE: This is not a homework question. I will be quite happy if some body can ...
4
votes
2answers
202 views

$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?

Let $f: \mathbb R \rightarrow \mathbb R $ be a continuous function such that $\int_{0}^{\infty} \,f(x) dx$ exists. Then Prove that incase (i) $f$ is a non negative function, then ...
1
vote
3answers
74 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
0
votes
1answer
86 views

how to integrate $\int \frac{\sin x}{x}$ in $[0,1]$ [closed]

I would appreciate if somebody could help me with this problem $$\int_{0}^{1}\frac{\sin{x}}{x}dx$$ here using Taylor series I got $\sum_{0}^{\infty} $$\frac{(-1)^n}{(2n+1)!(2n+1)} $ then what to do? ...
2
votes
1answer
69 views

Integration limits of the double integral after conversion to the polar coordinates

I want to solve the following double integral: $$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$ And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and ...
0
votes
0answers
48 views

Are the solutions to this integral known?

Mathematica knows that the real part of $y$ in this integral: $$\int_0^{\infty } \frac{1}{x^{1/y}+2} \, dx$$ is: $$0<\Re(y)<1$$ Therefore I am wondering if the solutions to this integral known? ...
3
votes
1answer
69 views

A possible closed form?

Does it have a closed form? $$\int _{0}^{\infty }\!{\frac {x\cos \left( x \right) -\sin \left( x \right) }{{x} \left( {{\rm e}^{x}}-1 \right) }}{dx}$$ EDIT: no need for answer, I just found the ...
7
votes
2answers
103 views

Convergence of $\int_0^1 \sin\frac{1}{x} \;\mathrm{dx}$

Given the following improper integral: $$\int_0^1 \sin\frac{1}{x} \;\mathrm{dx}$$ I know it converges, after substituting $u=\frac{1}{x}$ and then comparing to $\frac{1}{x^2}$ . But, is it also ...
6
votes
7answers
390 views

Integrating $\int_0^\infty\frac{1}{1+x^6}dx$

$$I=\int_0^\infty\frac{1}{1+x^6}dx$$ How do I evaluate this?
2
votes
3answers
133 views

Compute $\int_0^\infty\frac{\cos(xt)}{1+t^2}dt$

Let $x \in \mathbb R$ Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ . Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher ...
2
votes
1answer
49 views

Numerical integration of innocent-looking singular integrand

Consider the rather innocent integral: $$I=\int_{0}^{1}a x^{a-1}dx=1,\quad 0<a<1$$ Numerically, this integral converges awfully slowly, and one must use a recursive method to get anywhere near ...
2
votes
1answer
92 views

Solving Integral that contain exponential and Power

I have an integral of this form: $$\int_0^\infty e^{-\frac{x}{a}-\frac{z^2}{bx}-\frac{z}{bx}}\left(\frac{c}{c+x+z}\right)^K~dx$$ where $K$ is a positive integer. $a$ , $b$ and $c$ are reals and ...
3
votes
2answers
85 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
2
votes
1answer
41 views

A question in integral convergence

For what real values of $a$ the following integral converges? $$ \int_0^1 (-\ln{x})^a dx $$
1
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1answer
60 views

exponetial integral and tational

I have an integral with the form $$\int_b^\infty\frac{e^{-kt^2}e^{ct}t}{\sqrt{t^2-b^2}}dt$$ it is possible to find it analytically? thank you in advance
6
votes
1answer
50 views

convergence of $\int _1^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$

Question: $$\int _1^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$$ does this converge? Wolfram|Alpha doesn't have an answer, and I would really know. We tried to use Dirichlet and substituting with ...
2
votes
1answer
54 views

Integration of complex function

how do we do start doing the following integral? $$\int_{-\infty}^{\infty}e ^{-\pi(x^2a^2+2iux)}\,dx$$ Any help will be appreciated NOTE: $i$ is the square root of $-1$.
1
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0answers
47 views

Improper Riemann integral and imaginary exponential of real polynomials

Let $P(x_1,\cdots,x_n)$ be a real polynomial of degree $\geq 2$. What are the conditions on $P$ so that $$ I_P:=\int_{\mathbb{R}^n} e^{iP(x)} dx $$ exists as an improper Riemann integral ? Already ...
0
votes
0answers
36 views

Does the following complex intergral have an analitical solution?

Need an analytical solution to the following equation if possible: $$\int^{+\infty }_{-\infty } e^{-{x^2\over2} - ib(ax - \text{ln}(x) )} dx. $$ Where $i$ is the imaginary unit, $a$ and $b$ are ...
2
votes
1answer
91 views

How to integrate $e^{-\frac{x^2}{4}}\cos(x)$

I have the following integral I'm trying to solve: $$\int_0^\infty e^{-\frac{x^2}{4}}\cos(x)\,\mathrm{d}x$$ how to solve this, please help me.
1
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4answers
75 views

Convergence of the improper integral $\int_{0}^{\pi/2}\tan^{p}(x) \; dx$

I need help solving this integral: for which values of the parameter $p$ is this improper integral convergent? $$\int_{0}^{\pi/2}\tan^{p}(x) \; dx$$ Thanks a lot!
1
vote
1answer
57 views

Finding $\int_{-\infty}^{\infty}e^{x^2}\,dx$

I was wondering if $\int_{-\infty}^{\infty}e^{x^2}\,dx$ is defined, I think it isn't. But I dunno how to start proving/disproving.
1
vote
2answers
230 views

The unbeatable $\int e^{1/\cos(x)} dx$ integral

Is there any way to express this in non-elementary functions? $$ \int e^{1/\cos(x)} dx$$ And/or to calculate this definite integrals? $$ \int_{-\pi/2}^{\pi/2} e^{1/\cos(x)} dx$$ $$ ...
2
votes
2answers
101 views

the solution for an integral including exp and rational term

I is possible to find the following integral: $$ \int_1^\infty \frac{x^m e^{-ax}}{\sqrt{x(x-1)}}\;dx $$ Thank you in advance
0
votes
3answers
80 views

Evaluate the integration?

Find: $$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(3x^2+2\sqrt{2}xy+3y^2)}\,dxdy$$ I have no idea how to solve this,I would be thankful, if someone help me to solve this Thank you.
5
votes
8answers
264 views

Estimate $\int_0^{\infty} 1/\sqrt{1+x^4} \mathrm{d}x$

I need an analytical estimation of the following integral: $$\int\limits_0^\infty \frac{{\mathrm{d} x}}{\sqrt{1 + x^4}}$$ It has a root in the denomenator -- so I can't make use of complex residues ...
3
votes
1answer
107 views

Integration with a Bessel function

I have an integral with the form $$\int_1^\infty \frac{e^{-ax}}{\sqrt{x-1}} I_0(b\sqrt{x}) dx$$ where $I_0$ is the modified bessel function and a, b are constants. Is it possible to find this ...
1
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2answers
41 views

Integral convergence question

Let $ f: [a,\infty ) \to \mathbb R $ be $ f\in C([a,\infty)) $, with a cycle $ T > 0 $, and let $ g: [a,\infty ) \to \mathbb R $ be a monotonic function, and $$ \lim_{x\to \infty} g(x) = 0$$ ...
3
votes
1answer
61 views

Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
1
vote
3answers
165 views

A question related to a convergence of an integral

Let $f$ be $f\in C([0,\infty ])$, such that $\lim_{x \to \infty} f(x) = L $. Calculate $$ \int _{0}^\infty \frac{f(x)-f(2x)}{x}dx $$ Help?
4
votes
4answers
210 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
3
votes
4answers
126 views

How come does integral of $\ln^2 x$ from $0$ to $4$ , converges?

Would someone please help my understanding how come integral of $\ln^2 x$ from $0$ to $4$, converges? Thank you
0
votes
1answer
42 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
3
votes
3answers
139 views

How to compute $\int_0^1\frac{t\ln t}{1+t^2}$ ?

How to compute the integral $$\int_0^1\frac{t\ln t}{1+t^2}\ ?$$ So Wolfram alpha says it is exactly $-\dfrac{\pi^2}{48}$ . I tried many substitutions without success, and partial integration as ...
0
votes
2answers
49 views

Integral calculus exp

I have the following integral $\int_0^\infty e^{-6t}~dt=\dfrac{1}{6}$ and I can't remember the properties of integrals or "$e$" to get the result. Can you, please, help me, with the explanation? ...
3
votes
3answers
97 views

Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
1
vote
1answer
75 views

Solving a definite integral

How can i find the value of definite integral $$\int_{0}^{\pi}\lfloor\cot x\rfloor dx$$ Here $\lfloor a\rfloor$ means greatest integer value of $a$. My doubt is that $\cot x$ will lie between negative ...
0
votes
1answer
40 views

necessary condition of improper integral

Does a function's limit when x goes to infinity must be zero for its integral to converge? I had proved in my homework that if the function is non-negative, it's not necessarily true. Now I read a ...
0
votes
1answer
56 views

Showing a limit exists

No solution please. I want to show that the following limit exists $$ \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^{T}f(x)\,\overline{g(x)}\,dx $$ where $f$ and $g$ are finite linear combinations of ...