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.
76,735 questions
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Prove that $2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$
I'd like to prove that
$$2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$
Ok, someone said that this holds, but I tried really ...
- 115
2
votes
1
answer
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Changing the order of integration in an iterated integral with a single varible function [closed]
I am trying to consider a double integral:
$$
\int_t^\infty \int_s^\infty f(r) dr ds <+\infty
$$
where $f:\mathbb{R} \to \mathbb{R}$ is a smooth function, but NOT a non-negaitive function. And the ...
- 47
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If $\varphi(f)$ is Riemann integrable for each $\varphi \in E^*$, then is $f$ Riemann integrable? [closed]
Let $I=[0,1]$, $E$ be a Banach space and $f:I \rightarrow E$ be a map.
Suppose that for every continuous functional $\varphi\in E^*$, the map $\varphi(f):I\rightarrow \mathbb{R}$ is Riemann integrable....
- 173
0
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1
answer
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Base of the line integration [closed]
Does line integral integrate over the projection of a 3d curve onto the x-y plane or over the 3d curve itself as the base of the integration?
Thanks
1
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0
answers
110
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How to integrate $\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$ analytically?
I'm trying to solve the integral
$$\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$$
I do know that a similar integral
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
- 13
3
votes
2
answers
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Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra
This problem appears in the book:
Linear Algebra and its applications - David C. Lay - Fourth Edition
It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18
$(4.7), \...
- 5,844
-1
votes
0
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How to setup bounds for triple integrals [closed]
Set up (do not evaluate) triple integrals in spherical coordinates in the orders dρdϕdθ and
dϕdρdθ to find the volume of the cube cut from the first octant by the planes x = 1, y = 1
and z = 1.
7
votes
5
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Prove $\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx$ for decreasing function
Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that
$$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$
This would be quite obvious if $f$ were continuous. But for non-...
- 5,070
5
votes
0
answers
156
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On $\iint_0^\infty\frac{x^{s-1}\,y^{-s}}{e^{x+y}+1}\frac{dxdy}{\cosh{x}}$
From the definition of $\eta(s)$ and $\beta(s)$:
$$
\begin{align}
{2^{1-s}\Gamma(s)\,\eta(s)}&={\int_0^\infty\frac{x^{s-1}}{\cosh{x}}\,\frac{dx}{e^x}} \\
{2\,\Gamma(1-s)\,\beta(1-s)}&={\...
- 5,252
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Showing $\int_0^a\left(f(x)-\frac12x\right)^2dx\leq\frac1{12}a^3$ for $f(x)\geq0$ satisfying $\left(\int_0^tf(x)dx\right)^2\geq\int_0^tf^3(x)dx$ [duplicate]
Problem:
Given positive value $a$, we have $f(x) \geq 0,\forall x\in[0, a]$, and
$$\left(\int_0^t f(x) dx\right)^2 \geq \int_0^t f^3(x)dx, \quad\forall t \in [0, a]$$
Show that
$$\int_0^a \left(f(x)-\...
- 49
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0
answers
71
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Calculate $\int\cos(x)\ln(\cos(x))\,\mathrm{d}x$ [duplicate]
I was asked to calculate this integral
$$\int\cos(x)\ln(\cos(x))\,\mathrm{d}x$$
as part of my Real Analysis course.
I took the approach of integration by parts, denoting $u = \ln(\cos(x))$ and $v = \...
- 133
0
votes
0
answers
49
views
Decompose rational expressions [closed]
Partial fraction decomposition applies only when the degree of the numerator is less than the degree of the denominator.
A. True
B. False
It was in an exam. And the teacher answered A. But still, I ...
0
votes
1
answer
38
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Right continuity of $f$ and bounded variation on $[a,b]$ imples right continuity of $V(a,t)$
I'm studying from Bobrowski Functional analysis for probability and stochastic processes (not a university course). I got stuck on one of the exercises, exercise 1.3.4.
Let $f:[a,b]\to \mathbb{R}$ ([a,...
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1
answer
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Is this a probability density function? [closed]
Given $fx(x) = \{ \frac{1}{\pi} \; \text{for} \; x_1 + x_2 \le 1$
I am required to state if the function represents a density function and prove why. I know that to prove it I must check that $f(x) \...
5
votes
1
answer
288
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The Multiplicative Role of $dx$ in Indefinite and Definite Integrals: A Comparison with Derivative Notation
I understand from prior discussions (e.g., What does the $dx$ mean in the notation for the indefinite integral?) that $dx$ in $\int f(x) \, dx$ serves as more than mere notation for the variable of ...
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