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.
18 questions from the last 7 days
7
votes
5
answers
389
views
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
1
answer
284
views
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 ...
- 321
5
votes
0
answers
154
views
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 $\beta(s)$:
$$
\begin{align}
{2^{1-s}\Gamma(s)\,\eta(s)}&={\int_0^\infty\frac{x^{s-1}}{\cosh{x}}\,\frac{dx}{e^x}} \\
{2\,\Gamma(1-s)\,\beta(1-s)}&={\...
- 5,232
3
votes
0
answers
95
views
Showing $\int_0^a\left(f(x)-\frac12x\right)^2dx\leq\frac1{12}a^3$ for $f(x)\geq0$ satisfying $\left(\int_0^tf(x)dx\right)^2\geq\int_0^tf^3(x)dx$ [duplicate]
Problem:
Given positive value $a$, we have $f(x) \geq 0,\forall x\in[0, a]$, and
$$\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
1
vote
0
answers
106
views
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
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
- 13
3
votes
1
answer
86
views
Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra
This problem appears in the book:
Linear Algebra and its applications - David C. Lay - Fourth Edition
It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18
$(4.7), \...
- 5,844
0
votes
0
answers
58
views
Swapping sum and integral with an infinite series of modified Bessel functions
I am studying the following integral
\begin{align}
\int_0^{\infty} I_1(\sqrt{x}) \sum_{k=1}^{\infty} \Big( k K_1(k\sqrt{x}) - k^2 \sqrt{x}\, K_0(k\sqrt{x})
\end{align}
where $I_1$ and $K_\nu$ are ...
- 305
2
votes
0
answers
52
views
Inverse Fourier transform for the square root of the Ohmic bath spectral function
I think this is a bit hopeless but let me ask just in case. Consider the real and positive function:
$$
\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...
- 619
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 ...
-6
votes
1
answer
60
views
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) \...
0
votes
0
answers
71
views
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
1
answer
38
views
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,...
- 11
0
votes
1
answer
31
views
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
votes
0
answers
24
views
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.
0
votes
0
answers
13
views
If $\varphi(f)$ is Riemann integrable for each $\varphi \in E^*$, then is $f$ Riemann integrable?
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....
- 157