Questions about the evaluation of specific definite integrals.

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4
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3answers
917 views

Integral involving an error function

For $\sigma>0$, how can we prove that $$\frac{1}{2}\int_{-1}^1 \text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right) \, \mathrm{d}x= \text{erf}\left(\frac{\sigma}{2}\right)$$ where erf ...
0
votes
0answers
28 views

Changing limits of an integral

The integral: $$ \int_{-\infty}^0\frac{e^{i\alpha x}\,dx}{1+x+x^2}. $$ If I want to change the limits of this integral so that the integral is taken from $0$ to infinity instead of minus infinity to ...
3
votes
1answer
58 views

Evaluate a limit using integral

$$\lim _{n\to \infty \:}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2n-1}-\frac{1}{2n}\right)$$ How I evaluate this limit?
1
vote
2answers
36 views

Prove the following equation(much needed help):

Let there be a given function $f \in C([0,1])$, $f(x)>0$; $x\in [0,1]$. Prove $$\lim_{n\to\infty} \sqrt[n]{f\left({1\over n}\right)f\left({2\over n}\right)\cdots f\left({n\over ...
-4
votes
1answer
58 views

Does $\int_{0}^{1} |f(t)|dt=0$ imply that $f(t)=0\ \text{for all } t \in [0,1]$? [on hold]

$$\int_{0}^{1} |f(t)|dt=0$$ Does this equation imply that $f(t)=0$ for every $t \in [0,1]$? I need a proof of whether the answer is yes or no. I couldn't prove this; I started thinking, if it were ...
0
votes
2answers
59 views

Integral of series expansion…

Why is $$\int _0^1\:\left(1-x+x^2-x^3+x^4-....-x^{2n-1}\right)dx = \int _0^1\:\left(\frac{1-x^{2n}}{1+x}\right)dx,$$ why do we have $1+x$ in the denominator, and why does $1-x^{2n}$ appear in the ...
0
votes
3answers
22 views

To calculate the expectation of a variable with a given pdf

Let the pdf of a random variable X be given by $f(x)=ae^{-x^2-bx}, -\infty<x<\infty$. If $E(X)=-\frac{1}{2}$, then (A)$a=\frac{1}{\sqrt{\pi}}e^{-1/4},b=1$ ...
2
votes
1answer
58 views

Find the integral $\int_{-2}^{1} |x| d |x|$

To handle the integral $\int_{-2}^{1} |x| d |x|$ I do not see a way to begin with. If instead I am to compute $\int_{-2}^{1} |x| dx$ then things get easier, for simply computing $\int_{0}^{1} x dx + ...
6
votes
2answers
53 views

Help to resolve a Double Integral

I'm doing a workout guide about double integrals and I came across an exercise that I could not resolve for a while. $$\int_0^2\int_1^2 \frac{x}{\sqrt{1+x^2+y^2}} \,\mathrm dx\,\mathrm dy$$ I guess ...
-1
votes
1answer
66 views

A Logarithm Integral II [on hold]

Does the integral \begin{align} \int_{0}^{1} (1-t)^{2} \, \ln^{k}(1-t) \, \ln^{m}(t) \, dt \end{align} have a compact form for $m = 1$, and $m=2$ ?
2
votes
4answers
64 views

Evaluate $\int_{0}^{1}(1-x)^ndx$ by expanding the bracket.

I'd like to get a hint on this exercise. I believe I'm somewhat close to the answer. I used the binomial theorem to get: $\displaystyle\int_{0}^{1}(1-x)^ndx = \int_{0}^{1}\sum_{k=0}^{n}{n\choose ...
1
vote
1answer
36 views

Prove $\int^a_0 x(a^2-x^2)^{\nu-1} I_0(b x)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma(\nu)I_\nu(a b)$

How can I prove the following equality? $$ \int^a_0 x\left(a^2-x^2\right)^{\nu-1} I_0\left(b x\right)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma\left(\nu\right)I_\nu\left(a b\right), $$ under the ...
2
votes
1answer
68 views

Closed form for an almost-elliptic integral

Does $$\int_0^{2 \pi} \log\left(\frac{1}{2}[1+\sqrt{1-(a \sin\phi)^2}]\right) d\phi $$ have a closed form ? An approximation for small $a$ is $2E-\pi$, but it is the exact form that is needed ...
1
vote
1answer
23 views

Rearranging double integral and bounds

I am trying to figure out why we can rewrite $\int_0^n s (\int_0^s 1 \, dt) \, ds = \frac{n^3}{3}$ as $\int_0^n 1 (\int_s^n t \, dt) \, ds = \frac{n^3}{3}$ I would appreciate any pushes in the ...
0
votes
1answer
21 views

Cartesian to Spherical Coordinate Conversion for Triple Integral

I have a question regarding what happens to the boundaries when converting a triple integral from Cartesian to Spherical Coordinates. Example ...
1
vote
0answers
20 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
0
votes
1answer
43 views

A Trig Integral

Does the integral \begin{align} \int_{0}^{\pi/2} \cos(x) \, \ln\left( \frac{1 + a^{2} \sin(x)}{1 - a^{2} \sin(x)} \right) \, dx \end{align} have a closed form and what is changed if the limits are ...
0
votes
2answers
24 views

Monotone & bounded of an integral function .

Let , $f: [0,1]\to \mathbb R$ be such that $f(t)\ge 0$ for all $t\in [0,1]$. Define , $$g(x)=\int_0^xf(t) \,dt.$$ Then (A) $g$ is monotone & bounded (B) $g$ is monotone but not bounded (C) $g$ ...
1
vote
1answer
31 views

Integrating the log-normal function

Compute $$F(t)=\int_0^t \frac{1}{\sqrt{2\pi}\sigma t} \exp\left[-\frac{1}{2}\left(\frac{\log t-\mu}{\sigma}\right)^2\right]\,dt; t>0$$ My Attempt: $u=\frac{1}{t}\Rightarrow du=-\frac{1}{t^2}dt$ ...
1
vote
1answer
40 views

Integral involving $\operatorname{sinc}$ and exponential

Is there a closed form for the following integral: $$\int_{0}^a\exp\left[\frac{i\pi x^2}{b}\right]\operatorname{sinc}\left(\frac{\pi ax}{b}\right)dx$$ where $i=\sqrt{-1}$ and ...
2
votes
3answers
114 views

Prove $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for smooth $g$ with $g(0)=g(1)=0$ [on hold]

This came up in an optimization problem. How do you prove that $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for any $g$ which is twice continuously differentiable on $[0,1]$ and such that $g(0)=g(1)=0$?
0
votes
2answers
18 views

double integral question involving volume

find the volume of the solid with height h(x, y)=xy and base D where D is bounded by y=x+2and y=x^2. I believe this is a double integral question. I'm really not sure how to set up the bounds of ...
2
votes
0answers
26 views

Why do we set $x+1=\frac{1}{t}$ When we compute $I= \int\limits_{\frac{1}{2}}^{0} \dfrac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$

When we compute $I=\displaystyle \int\limits_{\frac{1}{2}}^{0} \dfrac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$ We set $x+1=\dfrac{1}{t}$ and we have $\displaystyle I=\int\limits_1^2\dfrac{dt}{\sqrt{4t-1}}$. I ...
2
votes
1answer
50 views

A definite integral contianing ln(x)

everyone, I met a tough definite integral as follows, $$I = \int\limits_1^\infty {\frac{{\ln x}}{{{{\left( {x + a} \right)}^m}{{\left( {x + b} \right)}^{n + 1}}}}} dx,$$ where $a$ and $b$ are ...
3
votes
2answers
99 views

How to solve:$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
0
votes
0answers
16 views

How to find a sum for power series in a given interval [on hold]

for this power series, first thing I tried to do was try to take integrals term by term and then find the definite integral in the given interval this is the integral that I find, but here is my ...
0
votes
1answer
21 views

Volume of Unusual Shape

Let $z = 1/r^n$, where $r = \sqrt{x^2+y^2}$ and $0 < n < 2.$ Note that this function has a discontinuity at the origin. Find the volume, $V(a)$, under this surface (and above the xy-plane) ...
1
vote
3answers
335 views

Very Tricky Double Integral Problem

Evaluate the Integral: \begin{align} \int_{0}^{4}\int_{\sqrt(x)}^{2}1/(1+y^3)\ dy dx\\ \end{align} I can't understand how this would be possible. There IS a formula for evaluating $1/(1+y^3)$, but ...
4
votes
2answers
193 views

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

Let $m,n,k$ be nonnegative integers. How might I go about evaluating the following integral? $$ \int_{-\infty}^\infty \left( \frac{\mathrm{d}^m}{\mathrm{d}x^m} e^{-x^2} \right) \left( ...
0
votes
0answers
32 views

Definite integral of Bessel function product over fourth power

This is an integral from Anders Losberg in 1960 related to concrete pavement theory: $$\int_0^\infty\frac{J_0(\alpha r)J_1(\alpha a)}{\alpha(1+\alpha^3\ell_e^3)}d\alpha$$ I'm wondering how to solve ...
0
votes
1answer
20 views

Integrating over 3 surfaces of a tetrahedron

I am supposed to integrate over the surfaces of a tetrahedron to find $\int \int curl(yi+2j)\cdot n{\partial p}$ where ${\partial p}$ is the surface of the tetrahedron bounded in the first octant by ...
0
votes
1answer
46 views

Having difficulty in evaluating a definite integral by substitution.

The question is this - $$I =\int_{-1}^1 \frac{e^{1/x}}{x^2(1+e^{2/x})}dx $$ So I started by substituting $\frac{1}{x}=t \rightarrow \frac{-1}{x^2}dx=dt. $ $$I=-\int_{-1}^1 \frac{e^t}{1+e^{2t}}dt$$ ...
3
votes
2answers
54 views

How to compute the integral $\int_{0}^{\pi} (\sin \varphi)^{5 /3} d\varphi$?

How to compute the integral $\int_{0}^{\pi} (\sin \varphi)^{5 /3} d\varphi$? I tried to use $(\sin \varphi)^{5 /3} = (\sin \varphi)^{2} (\sin \varphi)^{-1/3}$. But it is not successful. Thank you very ...
0
votes
1answer
14 views

Double integral of an rotated and translated ellipse

I need to solve this double integral, but can't find the especific change of coordinates. The region is an ellipse with rotation and translation of axis, I think that I need to find out the angle of ...
1
vote
2answers
44 views

Integrals with the special functions $Ci(x)$ and $erf(x)$

I'm looking for the solutions of the following two integrals: $$I_1=\int\limits_0^\infty dx\, e^{-x^2}Ci(ax)$$ and $$I_2=\int\limits_0^\infty dx\, e^{-ax}erf(x)$$ with ...
4
votes
1answer
43 views

Definite integral of inverse function

Task is: $$\int_{\sqrt(\pi/6)}^{\sqrt(\pi/3)}\sin(x^2)dx + \int_{1/2}^{\sqrt{3}/2}\sqrt{\arcsin x}dx$$ If we say, that $f(x) = \sin(x^2)$, and $a = \sqrt(\pi/6)$, $b= \sqrt(\pi/3)$ then we have: ...
1
vote
1answer
48 views

Calculate the integral $\int_{[0,1]\times[0,1]}\frac{1}{(1-xy)^a}dydx$ for $a>0$

I'm having trouble with the integral $\int_{[0,1]\times[0,1]}\frac{1}{(1-xy)^a}dydx$ for the case in which $a\neq{1}$. For the case in which $a=1$ it was not difficult to calculate the integral and I ...
11
votes
2answers
114 views

Closed form for ${\large\int}_0^\infty\frac{x\,\sqrt{e^x-1}}{1-2\cosh x}\,dx$

I was able to calculate $$\int_0^\infty\frac{\sqrt{e^x-1}}{1-2\cosh x}\,dx=-\frac\pi{\sqrt3}.$$ It turns out the integrand even has an elementary antiderivative (see here). Now I'm interested in a ...
0
votes
0answers
15 views

Question about application of double integrals to find out volume of solid cone!

A solid cone is obtained by connecting every point of a plane region S with a vertex not in the plane of S. Let A denote the area of S, and let h denote the altitude of the cone. Prove that the ...
0
votes
0answers
19 views

Question about reversing the order of integration (picture included!)

I want to reverse the order of integration. I think region over which we are integrating is like that (as shown in picture). So, I split it into two cases. First is region from 0 to 1 and second is ...
2
votes
0answers
17 views

If $X_1,X_2$ are independent beta then showing $\sqrt{X_1X_2}$ is beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
6
votes
1answer
105 views
+100

The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: ``For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
1
vote
1answer
17 views

The volume of the solid, generated by revolving about $y = 2$ the region bounded by $y^2\leq 2x, x \leq 8$ and $y \geq 2$, is

The volume of the solid, generated by revolving about $y = 2$ the region bounded by $y^2\leq 2x, x \leq 8$ and $y \geq 2$, is (A) $2\sqrt2\pi$ (B) $28\pi /3$ (C) $84\pi$ (D) none of these. My Steps: ...
2
votes
3answers
63 views

Integral of $\log(\sin(x)) \tan(x)$

I would like to see a direct proof of the integral $$\int_0^{\pi/2} \log(\sin(x)) \tan(x) \, \mathrm{d}x = -\frac{\pi^2}{24}.$$ I arrived at this integral while trying different ways to evaluate ...
-2
votes
1answer
36 views

The value of $\lim\limits_{n\to\infty}[(n+1)\int\limits_{0}^{1}x^n\ln{(1+x)}\;dx]$ is [closed]

The value of $\lim\limits_{n\to\infty}[(n+1)\displaystyle\int\limits_{0}^{1}x^n\ln{(1+x)}\;dx]$ is (A) $0$ (B) $\ln2$ (C) $\ln3$ (D) $\infty$ I need some hints to solve this. Please help. ...
2
votes
1answer
59 views

Show that $\int_{0}^{4}\sqrt{\frac{x}{4-x}}\arctan\left(\sqrt{\frac{x}{4-x}}\right)dx=2+\frac{\pi ^2}{2}$

Show that $$\int_{0}^{4}\sqrt{\frac{x}{4-x}}\arctan\left(\sqrt{\frac{x}{4-x}}\right)dx=2+\frac{\pi ^2}{2}$$ It is clear the function gives infinity value when the $x=4$, so how can I find the above ...
2
votes
0answers
16 views

Evaluate the following integral involving sign function

I = $\int_0^{x_1}\int_0^{x_1}\text{sgn}(y-x) x^{\alpha+i-1}e^{-x/2}y^{\alpha+j-1}e^{-y/2}dx dy$ where, $\text{sgn}(x) = 1$ if $x>0$ and $\text{sgn}(x) = -1$ if $x<0$. Otherwise it is zero.
1
vote
1answer
78 views

Is there a novel way to integrate this without using complex numbers?

I've been reading a post on Quora about lesser known techniques of integration and I'm just curious if there's also a novel way to integrate this type of integral without resorting to complex ...
7
votes
1answer
112 views

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

How to find the value of $$I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$$ If we put $$I_2=\int_0^\infty\frac{\arctan^2({x})\log({1+x^2})}{\sqrt{x}(1+x^2)}dx$$ After long ...
4
votes
0answers
40 views

Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...