Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
74,412
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Change of varibles in triple integral [closed]
P is the Parallelepiped with a Vertex in O(0,0,0) and it spread by 3 vectors.
V1=(2,0,1) , V2= (0,1,0) , V3=(1,-1,1). how to convert P into a cube to perform the triple integral.
∭P sin(π⋅(x−z))⋅e^(2⋅...
5
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2
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Can the differential be unitless while the variable have an unit in integration?
Apologies for terminology inconsistencies, as I'm reading a Chinese statistics and probabilities textbook while looking up intrinsics on an English encyclopedia.
This arose when I was reading the ...
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2
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Definite integral $\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx$?
For positive $a$ and $b$, consider the integral
$$ I(a,b)=\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx. $$
Does $I(a,b)$ have a closed-form expression (far-fetched hope)? If not, does it ...
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Does U(f,P,[a,b]) = L(f,P,[a,b]) really imply f is constant
I am currently reading Measure, Integration & Real Analysis by Sheldon Axler, and am working through the practice problems. In particular, I am on this problem right now:
Suppose $f:[a,b]\to\...
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55
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Double integral of projectile problem
Given the projectile problem
$$x^{\prime \prime}=-\frac{1}{(1+\epsilon x)^{2}}$$
$$x(0) = 0 \quad x^{\prime}(0) = 1$$
By integrating twice, the projectile problem can be rewritten as the integral ...
0
votes
0
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62
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Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$
Let $b-a=m$ and $b+a=n$, where $a$ and $b$ are real numbers. Then the following identities hold:
$$\frac{1-e^{\pm\text{i}\alpha}}{1+e^{\pm\text{i}\alpha}}=\tan{\left(\frac12\sec^{-1}\left(\frac{x+b}{a}...
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Why is the 6 dropped from the final integral solution?
I was looking at this website to find an integral solution.
But as you can see in the picture the final solution lacks the $6$ that's present above.
Is this an error or the $6$ should really be ...
- 115
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1
answer
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
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Integration of delta dirac to the power of N [closed]
I know that the Fourier transform of the Heaviside function is:
$ \int_{0}^{\infty} e^{-ikx} dx = \pi \delta(k)+1/(i k).$
Now I want to know, if I have an integral of a power of the right-hand side ...
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Theorem 7.39 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: How to extract $g$ from $G$?
Here is Theorem 7.39, in Chapter 7, in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:
If $f$ is continuous on the rectangle $[a, b] \times [c, ...
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Is the product integral positive?
Let $f$ be a smooth function on $\mathbb{S}^1$ with $\int_{\mathbb{S}^1}f(x)\mathrm{d}x=0$. Can the integral
$$\int_{0}^r\iint_{|x-y|<t}f(x)f(y)\mathrm{d}x\mathrm{d}y\mathrm{d}t$$
be nonnegative ...
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1
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Product of 2 normal variables with positive means
Suppose $X \sim N(\mu,1)$, $Y \sim N(\mu,1)$ are iid normal random variables with $\mu>0$. My research problem is finding out the asymptotics of the tail function of XY (since the explicit formula ...
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Integration by parts on an area
I'm reading an Engineering book. All I can think of is integration by parts
$$\int_{\Omega} \dfrac{\partial M_y}{\partial x}v\text{d}\Omega = M_y v|_{?}^{?} - \int_{\Omega} M_y\dfrac{\partial v}{\...
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Is $\frac{\cos(\theta)}{\Gamma(1+\alpha)}\int\limits_{0}^{\infty} \frac{t^{\alpha} (\cosh(t) - \sinh(t)) }{ 1- \cos(2\theta)e^{-2t} }dt \neq 0$?
$$
\mbox{Is}\ \cos(\theta)\int_{0}^{\infty}\!\frac{t^{\alpha} \left[\cosh(t) - \sinh(t)\right]}{ 1- \cos\left(2\theta\right){\rm e}^{-2t}}\,{\rm d}t > 0$$
for any $\theta \neq \pi/2 + \pi k$ and $\...
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Integrate $\int f(x) sn(x,m) dx$ where $f(x)$ is a polynomial
To find the antiderivative of a product of a polynomial $f(x)$ and a sine function we can use this formula:
$$\int f(x) sin(x) dx = F'(x) sin(x) - F(x) cos(x) + C $$
where $F(x) = f(x) - f''(x) + f^{(...
