Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
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How to evaluate the following exponential-trigonometric Integral?
How to evaluate the following Integral? $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{\cos2x}\sin (x+\sin2x)}{\sin x} \, dx$$
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What is $ \int_{0}^{\exp(-1)} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx $?
Background
According to p. 22 of the following paper by Blagouchine, we have the following Malmsten integral evaluation: $$ \int_{0}^{1} \frac{\ln \ln \frac{1}{x}}{1+x^{2}} dx = \frac{\pi}{2} \ln\left(...
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Integrate product of matrix exponentials of a symmetric matrix
Let $\mathbf{A}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible, symmetric matrix.
Let $\mathbf{Q}$ $\in \mathbb{R}^{N \times N}$ be a real, invertible matrix.
Given these properties of $\mathbf{...
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Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
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Double integral $ \iint_D (x^4-y^4) dx\,dy$
I have troubles with the following integral
$$
\iint_D (x^4-y^4) dx\,dy
$$
over D: $1<x^2-y^2<4, \sqrt{17}<x^2+y^2<5, x<0, y>0$
This is the same problem as in Compute $\iint_D (x^4-y^...
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Estimation of a gamma function-like integral
A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$
Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$
There are no conditions about $k$, so it can be non-integer.
...
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Moments of Pearcy type integral
In my research I encounter Moments of Pearcy Integral
which can be written as
$$
\int_{-\infty}^{\infty}x^{n}
{\rm e}^{-ax^{4} + bx^{2} + cx}\,{\rm d}x\qquad
a > ...
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Approximation of a Riemann sum.
Given a twice continuously differentiable function $f\in C^2([0,1])$, is there a theorem/result/algorithm on how to place $0<x_1<\ldots<x_{n-1}<1$ so that adding $x_0=0$ and $x_n=1$,
$$
\...
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$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$
Calculate:
$$\int_A xyz \ d \lambda_3$$
Solution:
We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$
...
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Calculate the surface integral of $F = \langle x,y,z\rangle$ over the surface given by $3x-4y+z=1$ [closed]
Calculate the surface integral of $F = \langle x,y,z\rangle$ over the surface given by $3x-4y+z=1$ for $0 \leq x \leq 1$ and $0 \leq y \leq1$, with an upward-pointing normal.
I'm not sure about how to ...
user1358219
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Will the following Method of engineering analysis work?
Analytical Engineering Analysis of 3D Shapes
Using volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a Analytical
Engineering Analysis of 3D Shapes without using mesh based FEA. Like
integration ...
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is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0 $?
when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞
\frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt $
and ...
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Formula for bump function
I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals:
$$
f(x) :=
\begin{cases}
0, & \mbox{if } x \le -1 \\
1, & \mbox{if } x = 0 \\
0, & \...
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How to evaluate $\int_0^{\infty } \frac{\sin (\pi x)}{\log (x)} \, dx$
I have tried different substitutions and transformations, but am not getting a lead. Any suggestion would be helpful.
The numerical value of the integral is around $-3.2192$.
Interestingly these two ...
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Variable transformation in the definite integral
recently I encounter a variable transformation problem in the derivation and I did not figure out how it works.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
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