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.
76,884 questions
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$I = \int_{0}^{\infty} \frac{1 - \cos x}{x(1 - e^x)} \, dx$
A nice question I found somewhere, solution using two approaches is given.
Any solutions using some other approach will be greater appreciated
- 125
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Computation of volume of $\mathrm{vol} (X_t) = 2^{r_1}\pi^{r_2} \frac{t^n}{n!}$ ( Jarvis's ANT book ) .
I am reading proof of Lemma 7.17 in the Jarvis's Algebraic number theory book and stuck at some statements.
Let $K$ be a number field with $[K : \mathbb{Q}] =n$. Then there are $r_1$ real embeddings ...
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Intuition behind substitution $x=\sqrt{\frac{a-t}{b+t}}$
Consider the following two integrals using Feynman’s trick.
(i) $I=\int_0^\frac{\pi}{2} \frac{\ln(1-\sin x)}{\sin x}dx$
$$I(t)=\int_0^\frac{\pi}{2} \frac{\ln(1-t\sin x)}{\sin x}dx $$
Using ...
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1
answer
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Is there an operator-theoretic derivation of $\int x^n e^{ax}\,dx$ using a finite Neumann series?
Consider the classical integral
$$
\int x^n e^{ax}\,dx,\qquad a\neq 0,\quad n\in\mathbb{N}.
$$
The standard method is repeated integration by parts. Since differentiation eventually annihilates the ...
- 121
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0
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60
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Convergence of the least-squares solution of a linear system
Suppose I have a linear system of the form
$$ A {\bf x} = {\bf b} $$
where the matrix $A$ and the vector $\bf b$ are functions of a continuous parameter $y \in ]y_{1}, y_{2}[$. I would like to find ...
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4
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1
answer
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A closed-form expression of $\int_0^\infty\frac{1}{x}\left(\sinh(a x)\text{sech}(2\alpha x)\text{sech}\left(\frac{\pi}{2}x\right)\cos(bx)\right)dx$
I would like to evaluate the following improper integral that arises in a fluid mechanics problem I am working on
$$
F=\int_0^\infty \frac{1}{x} \left( \sinh(a x) \operatorname{sech}(2\alpha x) \...
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4
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My Solution to $\int_{0}^{1}\ln(1-x)\ln(x)\ln(1+x)\,dx$ and solving a complicated related integral $I=\int_{0}^{1}\ln(1-x)\ln x\ln(1+x)\arctan(x)\,dx$
Before introducing the integral that I am currently trying to solve, I first wanted to share this rough solution (skipping a lot of steps) to the following related integral.
$$I=\int_{0}^{1}\ln(1-x)\...
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4
votes
1
answer
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Line integral around an arbitrary curve
In James Stewart's Early Transcendentals Ed. 9, at the end of the review for chapter 16, the following exercise is stated.
My struggle first came from finding that $\nabla \times \vec F=\vec 0$, so $\...
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1
answer
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Solve $I = \int_0^\infty \frac{dx}{(\pi^2 + (\ln x)^2)^2 (x + 1)^2}$ [duplicate]
Here's my approach
Let $x = e^u$, then $dx = e^u \, du$, hence -
$$I = \int_{-\infty}^\infty \frac{e^u \, du}{(\pi^2 + u^2)^2 (e^u + 1)^2}$$
Now, $\frac{e^u}{(e^u + 1)^2} = \frac{1}{(e^{u/2} + e^{-u/2}...
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votes
1
answer
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How should I integrate $\int Ce^{-3x}dy$ in this partial differential equation problem?
Solve the partial differential equation $3u_{y}+u_{xy}=0$. (Hint: Let $v=u_{y}.$)
Here's my work:
Consider the partial differential equation $3u_{y}+u_{xy}=0$.
Let $v=u_{y}$.
Then we have $v_{x}=u_{xy}...
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0
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what if the singularity of a function is a plane? [closed]
we know that the line integral is easily calculated by the residue theorem .
my question is :
"is there a residue theorem for the surfaces?"
if a multi valued function has a singularity for ...
6
votes
1
answer
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Is it valid to use the power series of $\ln(1+x)$ to compute $\int \ln(7x) dx$?
I tried to compute the integral
$$\int\ln(7x) dx$$
using the power series expansion of the logarithm.
My approach:
$$\ln(7x) = \ln(1 + (7x - 1))$$
Then I used the standard expansion:
$$\ln(1 + x) = \...
- 411
2
votes
2
answers
324
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How can this integral from a Romanian math magazine be evaluated?
I was browsing websites and social media when I came across a certain integral from a Romanian math magazine. The video presented the solution, and I tried to prove it.
$$ I = \int_{0}^{\frac{\pi}{2}} ...
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6
answers
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Looking for easier methods to evaluate $\int_0^1x^4\sqrt{4x^2+1}~\mathrm dx$
This integral arose while I was solving one of the questions of my Calc-II end-semester examination today:
Q. If $\gamma=\big\{(x,y)~\big|~x=y^2,y\in[-1,1]\big\}$ and $f(x,y)=xy$, compute $\...
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votes
3
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Evaluation of $\int_{0}^{\infty} \frac{dx}{bx^{4}+2ax^{2}+c}$
Compute the normalized form of the quartic integral $$\int_{0}^{\infty} \frac{dx}{bx^{4}+2ax^{2}+c}$$
I came across this problem from George Boros' book Irresistible Integrals. Is there a way to ...
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