Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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4answers
62 views

Integration with variable in numerator

$$\int\frac{x^5}{\sqrt{25-x^2}}dx$$ I tried to do it with substitution but couldn't get ride of $x^5$ in the numerator.
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2answers
90 views

integral problem - what is the quickest solution?

I am solving the below integral. $$\int_{0}^{1} (e^{\frac{-x}{a}}-a(1-e^{-\frac{1}{a}}))^2 dx$$ I can decompose the integrand to the simple elements doing all the algebra and then split and ...
2
votes
1answer
49 views

Trigonometric contour integral

I cannot figure out what I'm doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{...
1
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1answer
46 views

$\int \delta(x + xy/u - a)\delta(y + xy/v - b)f(x,y)dxdy$?

I need help evaluating the following integral: $$\int \delta(x + uxy - a)\delta(y + vxy - b)p(x,y)dxdy$$ where $\delta(x)$ is Dirac-delta function, and $p(x,y)$ is some sufficiently well behaved ...
0
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0answers
26 views

Can somebody check whether I have calculated this contour integral correctly?

$$\int_{|z-\frac{1}{2}|=1}\frac{e^{-iz}}{z(z-1)(z^2-1)} dz$$ I used the Residue Theorem and got this answer: $2i\pi-\pi e^i -\frac{3}{2}\pi i e^i$ Is there also some software that can compute these ...
9
votes
2answers
220 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
1
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1answer
40 views

Integral (log and exp)

Question (Someone asked me for help on this integral and I couldn't figure it out myself.) $$ \int_{-∞}^∞ log(1+ae^{-t^2})dt $$ Even taking the Taylor series such that $log(1+ae^{-t^2})$ ~ $...
0
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1answer
43 views

Improper integrals and convergence

Let C= {(x,y) such that x>0, y> 0}. Let f(x,y) = $\frac{1}{(x^2 +\sqrt x )(y^2 + \sqrt y)}$ Show that $\int_C{f}$ exists, do not attempt to calculate it. Attempt at at solution: I was thinking that ...
1
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3answers
84 views

Is $f(x)=\frac{1}{q}$ for $x=\frac{p}{q}$ and $f(x)=0$ else integrable?

Let $f:[0,1]\to\mathbb{R}$ be defined as: $$ f(x)=\frac{1}{q}\space\text{ if }\space x=\frac{p}{q}\space \text{ for some }\space p,q\in\mathbb{N} \space \text{ with } (p,q)=1\\ f(x)=0 \space\text{ ...
3
votes
3answers
78 views

Beginner Integration (Substitution)

I am pretty new to calculus and would like a nudge in the right direction in order to complete this question properly (Maybe also correct any misrepresentations I have about integration) So the ...
2
votes
3answers
143 views

How to integrate $\int (\tan x)^{1/ 6} \,\text{d}x$? [closed]

How do I compute the following integral $$ I=\int (\tan x)^{1/ 6} \,\text{d}x $$
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2answers
44 views

Evaluate$\int_0^{\infty}x^4e^{-3x}\; dx$ using the change of variable: $t=3x$

I have been given the hint that I need to use integration by parts more than once in order to get the answer. However, I can't seem to get a reasonable result.
1
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2answers
61 views

Evaluating the integral $\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx$.

How can I evaluate the following integral $$\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx.$$
0
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3answers
92 views

How does the integral $\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$ converge?

I tried using the fact that $\ln(f(x)) < f(x)$ but that doesn't seem to work. It's an improper integral. $$ \int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x $$
3
votes
0answers
75 views

How to integrate exponential and power function?

I am trying to solve the following integral $$\int_{0}^{\infty}e^{-(ax+bx^c)}\,dx ; ~~~a,b,c>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
3
votes
2answers
42 views

Calculate $\lim\limits_{R\rightarrow\infty}\int_0^\pi \cos(R\cos t)dt$ w.o. Bessel function

Im calculating a complex path integral to calculate $\int_0^\infty \frac{\sin x}{x}dx$. I was able to evaluate everything except the arc $\int_0^\pi i~\exp(iR~e^{it})dt$ where $R$ is the radius. I ...
1
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0answers
50 views

Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
3
votes
3answers
90 views

Why integration constant is real?

OK, we are all taught at school that the undefined integral of a function $f(x)$ is $$\int f(x)\;\text{d}x = F(x) + k$$ where $F'(x) = f(x)$ and $k \in \mathbb R$. But, why $k$ must be real? I know ...
2
votes
1answer
45 views

Evaluating the integral of a $\cos(\theta)$ within the exponential wrt $\theta$

I want to evaluate the following integral $\int^{2\pi}_0 \, d\theta e^{- i k (x - x')\cos{\theta}}$, where all of the variables are real and $i$ is the imaginary unit. The difficulty is the cosine ...
0
votes
0answers
34 views

Unclear passage in integration involving Gamma functions

I find myself in need of some advice on an integration problem. Let $F(x,\lambda)=\Gamma(x,\lambda)/\Gamma(x)$, $x,\lambda>0$ be the Regularized Upper Incomplete Gamma Function, where $\Gamma(x,\...
3
votes
1answer
28 views

Proving that $x^{\alpha}(1+\Vert x\Vert^{2})^{-k}$ belongs to $L^{2}(\mathbb{R}^{n})$

Let $\alpha\in\mathbb{N}^{n}$ be a multi-index, i.e. $\alpha=(\alpha_{1},\dots,\alpha_{n})$ such that $x^{\alpha}:=\prod_{i=1}^{n}x^{\alpha_{i}}_{i}$. The modulus of a multi-index is defined as the ...
4
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1answer
93 views

How to evaluate this Integral $\int { {\sqrt{5^2+K^2}}dK \over {\sqrt{10^2+K^2}K}} $

While working on an Exact Differential Equation, I encounter the following Integral. $$\int { {\sqrt{5^2+K^2}} \over {K\sqrt{10^2+K^2}}} dK$$ I have tried substitution and all the other elementary ...
1
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1answer
84 views

Can this integral be evaluated/approximated?

I've been trying to evaluate this integral without much success: $\displaystyle \int_{-\infty}^\infty dx\, e^{iax} \frac{1- e^{-c\sinh^2 bx}}{\sinh^2 bx}$ I've tried contour integration. There are no ...
1
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2answers
55 views

Integrals with limits

I am trying to do: $$\int_0^1 x\sin(180x^2)\, dx$$ I use substitution: Let $ u = 180x^2 $ and $ \tfrac{du}{dx}=360x $ $$ \implies du =360x \,dx $$ $$ \implies \frac{1}{360} \, du = dx $$ so we ...
1
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1answer
102 views

Substitution theorem for integrals, a “trick”

In Spivak's calculus, he shows the following example: $\displaystyle \int \dfrac{1+e^x}{1-e^x} dx$ and states that setting $u=e^x $and $du=e^x dx$ would work, even though $e^x$ is not there....yet. ...
3
votes
1answer
61 views

Would the order of Taylor Polynomial change after substitution?

I found the order of Taylor Polynomial is kind of confusing. For example, we know: $$T_4e^x = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \frac {x^4} {4!}$$ After substitute $x$ as $t^2$, we ...
1
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2answers
69 views

Suppose $f$ is continuous and verifies $f(x) = o(x)$. Does it follow that $\int_{0}^xf(y)\,dy = o(x^2)$?

I know $F(x) = \int_{0}^xf(y) dy$ means $F^\prime(x) = f(x)$, but I have no ideas how to relate it to little-oh test. The only method in my mind is to find example $f(y)$ and $F(x)$, but I don't ...
2
votes
4answers
139 views

How to show $\int_{1}^{\infty} \frac{\sin^2(x)}{x^2}dx$ is finite?

At first, my approach was to directly take the improper integral of it. However, it seems not that easy. Then I tried to find another fraction to make a comparison. I got $\frac{\sin^2(x)}{x^2} <...
1
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3answers
105 views

Integral Problem: $\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2}$

I am having trouble with this definite integral problem $$\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2} \, dx$$ I know that the solution is $$e - \dfrac{1}{e}$$ I checked the step by step solution from ...
1
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1answer
60 views

Related integral problem to the Gaussian integral

So according to Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$, $$\int_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ I want to solve for this. $$\int_0^\infty e^{-x^2}\ln(x)dx$$ ...
1
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1answer
30 views

Product of Hilbert bases of $L^2(\mathbb{R}^p)$ and $L^2(\mathbb{R}^q)$ is a Hilbert basis for $L^2(\mathbb{R}^{p+q})$

Let $(\alpha_n)_n$ be a Hilbert basis of $L^2(\mathbb{R}^p)$ and let $(\beta_k)_k$ be a Hilbert basis for $L^2(\mathbb{R}^q)$. I need to show that $(\alpha_n \beta_k)_{(n,k) \in \mathbb{Z}}$ is also a ...
2
votes
2answers
91 views

How do I evaluate $\int_{0}^{\infty} u^{z-1}(e^{iu}-1) \, du$?

I am trying to evaluate the following integral that shows up in this paper http://arxiv.org/pdf/1103.4306v1.pdf $I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$ for $-...
1
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0answers
63 views

The time taken for a particle to return to its initial position involving integration of $v(t)$

Here, for part (a), I have solved for $v(t)=0$ and arrived at the answer that $t=6$. However, for Part (b), the mark scheme given says to do it in the following way: I understood why the mark ...
33
votes
4answers
492 views

Integral whose upper limit is the integral itself: $\int_{0}^{\int_{0}^{\ldots}\frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$

I recently encountered the following definite integral: $$\int_0^{\int_0^\ldots \frac{1}{\sqrt{x}} \ \mathrm{d}x} \frac{1}{\sqrt{x}} \ \mathrm{d}x$$ where "$\ldots$" seems to indicate that the upper ...
1
vote
1answer
64 views

Definite or indefinite integration of a relationship in physics

When we have a relationship like the following $$V=L\frac{di}{dt}$$ then in order to find the current $i$ we have to integrate. My question is what how we choose what kind of integration we must ...
0
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1answer
64 views

Calculating the volume of a rectangular hole with sloping edges

I need some help calculating the volume of a hole. The bottom of the hole is a rectangle which will have the measurements 15 by 35 meters. The depth is 8 meters. Now here is the tricky part. All four ...
1
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1answer
75 views

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}$dx?

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $$\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}dx?$$ I just need to solve this integral. But, I don't know how. ...
1
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2answers
143 views

Formulae for Catalan's constant.

Some years ago, someone had shown me the formula (1). I have searched for its origin and for a proof. I wasn't able to get true origin of this formula but i was able to find out an elementary proof ...
2
votes
1answer
74 views

Absolute continuity and Radon-Nikodym derivative

Let $\nu$ be a measure and $\mu$ a finite measure on $(X,\Sigma)$ with $\nu \ll \mu$. (All $\mu$-null sets are $\nu$-null.) Theorem: There exists a measurable $f:X \to [0,\infty]$ such that $\...
1
vote
3answers
94 views

Integration of $\int_0^{+\infty} \frac{\ln(1/x)}{1-x^2}\,dx$

I am trying to integrate the below problem, but not sure if its integrable. $$\int_0^\infty \frac{\ln(1/x)}{1-x^2}\,dx$$ Had the denominator been $1+x^2$, I would have used tan(x) substitution. ...
0
votes
1answer
49 views

evaluate a Fraunhofer diffraction integral

I need to evaluate the following integral: $$\int_{-\infty}^\infty \text{rect}(\frac{x'}{2w}) \exp\left (\frac{im}{2}\sin \left(k_Gx' \right) \right )\exp\left (-\frac{ik}{z}xx' \right )dx'$$ where ...
0
votes
1answer
39 views

Convolution of two functions.

$f(x)=2x/3$, $0<x<3$, $f(x)=0$ otherwise $g(x)=1$, $-1<x<3$, $g(x)=0$ otherwise I am trying to work out the convolution $h=f*g= \int(f(y)g(x-y))dy$ I am able to show that: $x - 3 &...
4
votes
3answers
686 views

Confusion regarding $\log(x)$ and $\ln(x)$

I was solving an integral and I encountered in some question $$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$ I know its integration is $\log(x)$. But my answer comes correct when I use $\ln(...
1
vote
2answers
26 views

Seperation of variables algortithm.

I am doing a question for a past paper that asks me to solve using separation of variables: $y_{tt}$=$y_{xx}$-$y_x$+4y with Dirichlet zero conditions: y(0,t)=y(1,t)=0 and the initial data: y(x,0)=f(...
0
votes
3answers
63 views

Derivative of an Integral with Two Functions as Bounds

I've been asked to find the derivative of $$g(x)= \int_{\cos x}^{x^4}\sqrt{2-u} du$$ using the Fundamental Theorem of Calculus part 1, and I know I should be substituting and setting a variable to one ...
0
votes
1answer
53 views

Is this Complex Integration correct?

I want to integrate $\displaystyle \int_{-\infty}^\infty dx \, e^{iax}\frac{1-e^{-bx^2}}{x^2}$ for a>0. I am going to try and do this using the method of contour integration. I will choose a ...
1
vote
1answer
129 views

Change of variables into the unit ball

Question: Express the volume of the solid bounded in $\mathbb{R}^3$ bounded below by the surface $z = x^2 + 2y^2$, and above by the plane $z = 2x + 6y + 1$, as the integral of a suitable function over ...
6
votes
0answers
225 views

Seem to be encountering many interesting results from integrals in the form: $\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$

I have found interesting results from integrals in the form: $$I=\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$$ A few examples of interesting functions here are: $$f(a,x)=\sin(ax)\implies I=\...
2
votes
2answers
68 views

$\int_0^{a} x^\frac{1}{n}dx$ without antiderivative for $n>0$

My exercise is to find $\int_0^{a} x^\frac{1}{n}dx$ without antiderivatives for $n>0$. The first thing I did is plot of some of the $x^\frac{1}{n}$ for the first twenty n. This is what I got. ...
0
votes
3answers
46 views

How to test whether $e^{3x^{2}} + \frac {1}{1+3x^2} - 2\cos(x^2)$ is $o(x^3)$?

From what I learned, $\lim_{x \rightarrow 0} \frac {f(x)}{x^3} = 0$ tells $f(x) = o(x^3)$ In this case, I have tried to compute $\lim_{x \rightarrow 0} \frac {e^{3x^{2}}}{x^3}$, but the limit seems ...