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.
76,749 questions
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Find real functions $f$ such that $\forall n \in \mathbb{Z}^+, f(nx)=nf'(x)f^{n-1}(x)$
This question arose from the simple observation that if $f(x)=\sin(x)$
$$\sin(2x)=2\sin(x)\cos(x)=2f(x)f'(x)$$
However a similar property does not hold for $\sin(3x)$
This came with the additional ...
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Explicit bounds for polytope integral
In Theorem 1.1 of this paper a constant is defined by a massive polytope integral, which is subsequently evaluated for certain values of $r$ using a computer algebra system. My question is whether ...
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Wikipedia’s proof of the Laplace’s method of integration [closed]
The “upper bound” proof clause states the following:
Lastly, by our assumptions (assuming $a$, $b$ are finite) there exists an $\eta > 0$ such that if $|x - x_0| \geq \delta$, then $f(x) \leq f(...
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Check the convergence of the integration [closed]
Check the convergence of the following integration $I= \int_1^{+\infty}\dfrac{\cos x - \cos (2x)}{\sqrt{x}\ln (1+\sqrt{x})}$.
I hope to solve this problem and don't use Dirichlet' theorem. Thank you ...
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Simplify an expression for subsequent numerical integration
I have the following equation,
$$\epsilon = \frac{3A\int_{0}^{2\pi} p^2 d\phi}{2\int_{0}^{2\pi} p^2(p+\frac{1}{2} \sqrt{R^2-w^2\sin^2\theta}) d\phi}$$
where A, R, w and $\theta$ are constants for the ...
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Is this infinite series for $\pi$ already known? [duplicate]
I discovered this a few days ago when I was evaluating the integral $I=\int^{\frac{1}{2}}_{0}\sqrt{\frac{x}{1-x}}dx$. When substituting $x=\sin^2\theta$, $I=\frac{\pi}{4}-\frac{1}{2}$ but when ...
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Question on Stokes phenomenon in Picard-Lefschetz theory
I wish to ask a question on the Picard-Lefschetz method for computing conditionally convergent comlex integrals. There is a case in Picard-Lefschetz theory in which a steepest descent contour ...
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derivative of integral of a function with respect to the same function
While doing some calculation I came across the following term [$G=G(x,x'),$ $x$ is independent of $x'$] $$K=\frac{\partial G(x,x')}{\partial (\frac{\partial G}{\partial x'})}$$ I tried to think of it ...
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On the branch cut selection for evaluating a Gaussian integral with complex exponent
Consider the following integral:
$$I=\int_0^\infty dx\,e^{-x^2\frac{1+j}{\sqrt{2}}}.$$
where j is the imaginary unit.
We get:
$$I^2=\int_0^\infty \int_0^\infty dx dy e^{-(x^2+y^2)\frac{1+j}{\sqrt{2}}}....
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Getting different answers evaluating $ \int_{-1}^{1} \frac{\sin(\cot^{-1}x) + \cos(\tan^{-1}x)}{x^2 + 1}\,\mathrm{d}x$ with different identities
The integral is:
$$I = \int_{-1}^{1} \frac{\sin(\cot^{-1}x) + \cos(\tan^{-1}x)}{x^2 + 1}\,\mathrm{d}x$$
First method: We notice that the sine term is entirely
an odd function so its contribution to ...
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BC Calc AP weird question about integration involving points of a function [closed]
I need help with this problem. It starts with a graph that states when $x = 2$, $f(x) = 4$, and $f'(x)= 2$, and when $x=5$, $f(x) = 7$, and $f'(x) = 3$. The second part says if $\int_{2}^{5}f(x)dx=14$,...
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Ball inflation inequality on restriction theory, l^{2} decoupling
This year I have been studying the paper "The proof of $l^{2}$ decoupling conjecture" by Bourgain and Demeter. by Bourgain and Demeter. I have been able to understand every part of the proof ...
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For which functions $f$ is $\displaystyle \int_{t_0}^t f(s)\exp(s) ds$ solvable in elementary functions?
SKIP TO THE NEXT ALINEA TO SKIP THE CONTEXT........ To put a very long story short: this questions stems from my question earlier today. It comes from a question regarding how, or rather why, we "...
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Prove $\int (f - x/2)^2 \le a^3/12$ given $(\int f)^2 \ge \int f^3$ [duplicate]
I am working on a challenging integral inequality problem. I would appreciate a fully rigorous proof, especially concerning how to deduce the necessary pointwise constraints on the function $f(x)$ ...
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When curvature is proportional to inverse square of ordinate [closed]
After integration of (differentials primed w.r.t. $x$ )
$$ \frac{y''}{{(1+y^{'2})^\frac32}}=\frac{c_1}{y{^2}}$$
The general solutions seem to be the catenaries only
$$ y={c_1\cosh \frac{x}{c_1}}+c_2 $$...
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