All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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62 views

Evaluating an integral using logarithms.

Evaluate the integral: $$ \int \frac{1}{x\log_3 x} dx $$ I tried to change it to this form : $$ \int \frac{\ln 3}{x\ln x} dx $$ But i couldn't continue. How could i arrive to this form $ ...
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1answer
117 views

Find the Method Moment Estimator of parameter $\theta$

Find the MME of parameter $\theta$ in the distribution with the density $f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for $x>1$ and $\theta >0$. So far I think I have a basic understanding of the ...
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0answers
76 views

Validity of Substitutions in Integrals [duplicate]

In integrals, we often make 'u' substitutions, which involve writing $u=f(x)$, and then also writing $du=f'(x)dx$. How valid is this? I've heard that you aren't allowed to do things like 'break' up ...
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1answer
39 views

calculate the 3-integral

$\int\int\int\limits_{ V} z \, \mathrm{d}x \, \mathrm{d}y\, \mathrm{d}z $ $V=(x,y,z)\in R^{3}: \frac{x^{2}}{a^{2}} +\frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} \le1, z\ge0 $ I have to calculate the ...
2
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1answer
71 views

Why is $\int_0^x \int_0^u f(t)(u-t)dt \,du = \int_0^x\int_0^uf(t)(x-u)dt\,du$?

In this question I learned that $$\int_0^x \int_0^u f(t)(u-t)dt \,du = \int_0^x\int_0^uf(t)(x-u)dt\,du.$$ I've been trying to come up with an intuitive reason why this should be so, like a geometric ...
4
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8answers
278 views

Evaluating $\int \frac{1}{\sqrt{x^2 + a^2}}\, dx$ without resorting to trigonometric $u$-substitution

I am looking for a quick and intuitive way to evaluate this indefinite integral without resorting to any trigonometric functions. I'm not sure if it is at all possible to do so, but I was just ...
2
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0answers
39 views

comparison of two integrals

Let $n \in N$. How to compare two integrals: $$ I_1=\int_0^{\infty}\left(\frac{\sin t}{t}\right)^n dt \quad \text{and} \quad I_2=\int_0^{\pi}\left(\frac{\sin t}{t}\right)^n dt\,\, ? $$ I've beet ...
5
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4answers
523 views

The integral $\int \frac{\sin^3(x)}{\cos^4(x)}\,dx$

Hey I'm having problem solving this integral : $$\int \frac{\sin^3(x)}{\cos^4(x)}dx$$ I think we should use $t=\sin(x)$ but it's not working for me. and if I use $t=\tan(x/2)$ it gets worse. Any ...
2
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1answer
46 views

Integration by Parts-Riemann-Stieltjes integral

How would I integrate the following? $\int_a^\infty (w-a)dF(w)$ for any fixed $a$, where $F(0)=0$ and $F(w)$ is strictly increasing and converges to $1$ as $w\to \infty$ . I've started by using the ...
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1answer
33 views

what is integral of $\int(1/2)exp^{-|x|} dx $ for -m to m with m is real number

i have some difficulty to finish this integral, can you show me how to find the result of this integration? $$\int(1/2)e^{-|x|} dx $$ for (-$m$ to $m$) whdere $m$ is a real number.
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1answer
52 views

Most powerful size $\alpha$ test

Someone can help me to check this answer? How to find the Most Powerful Test size $\alpha$ and Power of Test, Since I have $H_0 : X \thicksim f_{\theta 0}= (1/\sqrt(2\pi) \exp^{(-x^2/2)}$ and $H_1 : ...
1
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1answer
56 views

Integration Problem with a particle moving along x-axis

How would you solve this problem? If we have a particle moving along the x-axis with acceleration $a(t) = 9*t^2 - 4*t - 8$, (time must be positive or 0). The particle is at rest when t = 0, located 7 ...
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1answer
83 views

Integral of a function with big power $\int\frac{dx}{\cos^8 x}$

I'm trying to integrate this function and I know that $\cos(x)$ in denominator with even power we should use z=tanx to solve the integral. But I'm not succeeding in solving it. Any ideas ? ...
1
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1answer
46 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
2
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3answers
244 views

Why is integration the inverse of differentiation

Why is integration the inverse of differentiation, I mean why do I get the same function when I integrate and then differentiate the result?
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1answer
126 views

Existence theorem for antiderivatives by Weierstrass approximation theorem

Is there a way of proving the existence of antiderivatives (of continuous functions on a compact subset of the real line) without using tools of integration? This is an exercise in: ...
2
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1answer
80 views

Application of Stokes' Theorem

Calculate the integral $\int_\Gamma(z-y)dx+(x-z)dy+(y-x)dz$ by using Stokes' Theorem. Where $\Gamma$ is surface obtained from intersection $x^2+y^2=1$ and $x+2z=1$. I don't know how to solve this ...
0
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1answer
93 views

integral from 0 to infinity of limiting function

If $$f(x) = \displaystyle\lim_{n\to\infty}{\cos(x)\over1+(\arctan(x))^n},$$ find integral the $\int_0^\infty f(x)\,dx$. I tried to put $\arctan(x)=t$ and transformed limits to 0 to $\pi/2$ and ...
11
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1answer
332 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} {\cos\left(\pi x^{2}\right)\over 1 + 2\cosh\left(\,2\,\pi\,x\,/\,\sqrt{\,3\,}\,\right)}\,{\rm d}x ...
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1answer
63 views

Explicit Equation Trouble of Integral

$$\int 1 dt=\int\frac{1}{\sin z-1}dz \\=\int\frac{1}{\sin z-1}.\frac{\sin z+1}{\sin z+1}dz \\=\int\frac{\sin z+1}{-\cos^2 z}dz$$ I need the explicit solution of this integral where $z = y - t$. Here ...
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3answers
64 views

Finding the derivative of a definite integral

$$ G(x)=\int_1^{x^2}(x-t)\sin^2(t)dt $$ Find $ G'(x) $ given $G(x)$. Normally I can solve these types of problems, but I'm thrown off by the two variables present, both $x$ and $t$ under the ...
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2answers
53 views

Show that $\int_x^1\frac{dt}{1+t^2}=\int_1^{1/x}\frac{dt}{1+t^2},\;x>0.$

Show that $$\int_x^1\frac{dt}{1+t^2}=\int_1^{1/x}\frac{dt}{1+t^2},\;x>0.$$ So, I'm learning about integration techniques, and I get this exercise. We've been practicing $u$-substitution, and I ...
4
votes
1answer
31 views

Show that the sequence $\Omega^0\Bbb{R}^2\ \longrightarrow\ \Omega^1\Bbb{R}^2\ \longrightarrow\ \Omega^2\Bbb{R}^2$ is exact.

I have been posed the following question, which I am unable to answer: Let $a_1,a_2\in\mathcal{C}^{\infty}(\Bbb{R}^2,\Bbb{R})$ be infinitely differentiable functions such that $\frac{\partial ...
1
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4answers
75 views

Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]

Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b]. *I'm thinking about this but without progress...
3
votes
3answers
125 views

How to calculate $\int \frac{x}{\sqrt x -2}dx$

I don't know how to solve the following integral. I need some suggestions. Thank you! $$ \int \frac{x}{\sqrt x -2}dx$$
3
votes
1answer
54 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
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0answers
114 views

Gauss Hermite quadrature on finite interval

I would like to approximate an Integral of the type $I = \int_{x_l}^{x_u} f(x) w(x) dx$ where $w(x) = \frac{1}{2\pi}e^{-\frac{1}{2}x^2}$ and $f(x)$ is only defined on the Intervall $D = [x_l, \, ...
1
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1answer
66 views

Computing the moment-generating function of $f(x)=e^{-x}$

I have the following in my textbook: And I'm trying to verify that $M(t) = (1-t)^{-1}$ on my own. I'm getting: $$ f(x) = e^{-x} $$ $$ M(t) = \int_0^\infty e^{tx}f(x) ~dx $$ $$ = \int_0^\infty ...
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1answer
88 views

Leibniz rule in a Double Integral

I have been trying to evaluate the following double integral: $$\frac{\partial}{\partial \theta_1 \partial \theta_2} \int_{\theta_1-\theta_2}^{\theta_1+\theta_2} \int_{\theta_1 -\theta_2}^{x} u(y,x) ...
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4answers
115 views

How to solve $\displaystyle \int \frac{e^{2x}+e^{-2x}}{e^{2x}-e^{-2x}}dx$?

I don't know how to solve the following integral. I need some suggestions. Thank you! $\displaystyle \int \frac{e^{2x}+e^{-2x}}{e^{2x}-e^{-2x}}dx$ (I need to use some kind of a substitution of the ...
1
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2answers
116 views

Explain me definite integral…

So, as the derivative is a tangent of an angle between the $x$-axis and the corresponding tangent line, how can we represent an indefinite integral, and why is the area (definite integral) of, for ...
2
votes
1answer
235 views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
6
votes
3answers
889 views

integral from zero to zero

it seems obvious that this integral is zero and so is the limit but what theorem we are using here? I see it's connected to Riemann sums with an interval=zero Right ? The function $\mathrm{f}$ is ...
1
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1answer
78 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
4
votes
2answers
265 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
77 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 ?
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3answers
82 views

Integral $\int_{0}^{1}\frac{\sin{x}}{x}\operatorname d\!x$

In this problem $$\int\limits_{x=0}^{x=1}\frac{\sin{x}}{x}\operatorname d\!x$$ using Taylor series I got $\sum_{0}^{\infty} $$\frac{(-1)^n}{(2n+1)!(2n+1)} $ then what to do? Is it the final ...
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1answer
48 views

Constructing a weighting function with equal mean on two random variables

I am not a mathematician, but I hope that it is understandable. I try to tackle a problem which can be described as the following: Let $X_1$ and $X_2$ be random variables with same support $\Omega$ ...
1
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1answer
40 views

Linearity of the Riemann integral with complex valued functions

I want to verify the following facts for Complex integrals assuming that this works for real integrals: Let $f,g:[a,b] \to \mathbb{C}$ and $a \in \mathbb{C}$. (a) ...
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2answers
44 views

Multivariable Calculus Length of Curve

I have to find the length of a curve C which is parametrized by $x(t)=\dfrac{e^t+e^{-t}}{2},$ $y(t)=\cos(t)$ and $z(t)=\sin(t)$ where $t$ goes from -1 to 5. I believe this involves simply finding the ...
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1answer
61 views

Have no idea to evaluate this integral with e

Evaluate $\int\frac{e^{2x}}{4+e^{4x}}dx$ Have no idea to evaluate this integral with e
0
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1answer
54 views

Changing bounds with integration using $u$ substitution

I am trying to find the value of the following integral: \begin{equation} \int_0^5 x\sqrt{25 - x^2} dx \end{equation} I know that $u$ would be equal to $25-x^2$ and $du$ would equal $-2xdx$. Then ...
3
votes
1answer
30 views

Finding transformed region by change of variables

I have the equation of a curve $x^{2/3} + y^{2/3} = a^{2/3}$ and I'm using the change of variables $x = u\cos^3v $, $y = u\sin^3v$ . I have calculated the Jacobian ...
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3answers
54 views

Show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $

Hello everyone I have been trying to show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $ but I keep ending up with something like this: $2d{\sin(\frac{x}{2})}\cos(\frac{x}{2}) $ ...
3
votes
4answers
207 views

evaluating $\int x^2\sqrt{x^2+16}\;\mathrm{d}x$

How do I find $$\int x^2\sqrt{x^2+16} \;\mathrm{d}x$$ I tried a tangent-based substitution, but it didn't seem to work.
4
votes
0answers
95 views

$\int e^{-x} \log \log x dx$ - another special integral?

I came across this integral in some old notes. After several unsuccessful attempts I ran it in WA and got an interesting result: the antiderivative (closed form) doesn't exist, but the bounded ...
0
votes
1answer
369 views

Find the volume of the solid obtained by rotating the region enclosed by the curves $y=x^2 , x = 1, x = 2$, and $y=0$ about the line $x=5$

Find the volume of the solid obtained by rotating the region enclosed by the curves $y=x^2, x = 1, x = 2$, and $y=0$ about the line $x=5$. I set up the question using the cylindrical shells method. ...
2
votes
3answers
140 views

Particular integral of $\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 4y = \mathrm{e}^x\ $

I need to find the particular integral for the following equation: $\dfrac{d^2y}{dx^2} - 5\dfrac{dy}{dx} + 4y = \mathrm{e}^x\ $ So far I have found that $y = A\mathrm{e}^{4x}+B\mathrm{e}^x $. Then ...
0
votes
1answer
52 views

Prove that $x \rightarrow \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$ is convex

To put it bluntly I'm stuck proving proving the subsequent inequality $$ \forall x>0, \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt \int_0^\infty \frac{t^2 e^{-tx}}{1+e^{-t}}dt \geq {\left ( ...
8
votes
1answer
130 views

Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...