Questions tagged [integration]
Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.
56,519
questions
0
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0answers
16 views
numerical integration of a Differential Equation in C++ [closed]
I need to find a definite integral of an equation in C++.
dy/dt=(v/L)sin(y +(v/L)*ln[cos(Kt+a)] - b)
v, L, K, a, b are constants, 'y' is a function of 't'.
Now, I need to find the integral of 'y' ...
1
vote
1answer
25 views
How to show that $J_{n+1} = \frac{3n-1}{3n} J_n$?
Let $$J_n := \int_{0}^{\infty} \frac{1}{(x^3 + 1)^n} \, {\rm d} x$$
where $n > 2$ is integer. How to show that $J_{n+1} = \frac{3n-1}{3n} J_n$?
1
vote
0answers
12 views
Change of variables in integrals
Im trying to understand what kind of variable change in integrals is legit.
In my book's it states that the function the deriviative of the function that we change should be continious, but I've seen ...
0
votes
4answers
59 views
Find $\int\,\frac 4 { x \sqrt{x+1} }\,\text{d}x$
Calculate
$$ \int \,\frac 4 { x \sqrt{x+1} } \,\text{d}x\,.$$
My attempt: I tried substituting $u=\sqrt{x+1}$, so $u^2=x+1$, but it didn't get me anywhere.
1
vote
0answers
12 views
What is the functional derivative of a function depends on derivative at boundary
I have the following problem:
There is a functional $I[f]$:
$$I[f]=\int_{-1}^1\Bigl(\frac{\mathrm{d}f}{\mathrm{d}y}_{|y=-1}-\frac{\mathrm{d}f}{\mathrm{d}y}_{|y=1}\Bigr)f\mathrm{d}y$$
where $\frac{\...
0
votes
2answers
28 views
Compute pdf parameters given a probability value
Suppose I have a certain distribution i do not know the parameters. As an example, consider a normal distribution $\mathcal{N}$ with mean $\mu$ and variance $\sigma^2$. Given $X \sim \mathcal{N}(\mu,\...
0
votes
2answers
35 views
integration's extremes
I apologize in advance for the question. I have $\int_{-1}^{1}xe^{-\frac{x^2}{2}} dx =\int_{...}^{...}xe^{-t}\frac{dt}{x}$ but…
if $x=-1\Rightarrow t=\frac{(-1)^2}{2}=\frac{1}{2}$
if $x=1\Rightarrow ...
0
votes
3answers
24 views
False statement about a continuous and non negative function
It was asked in University of Hyderabad exam(2017).
I have shown 2nd and 3rd Statments to be true with the help of "Intermediate Value Theorem for Integrals".
But i can't reason why the ...
0
votes
0answers
14 views
Measurement limitations in definite integrals over continuous variables
This is a theoretical issue I found myself thinking about:
$\int p(x)dx$
is an integral of a pdf for random variable $X$,
and $\theta$ is a threshold point on that pdf
It's clear that $P(x=\theta)$=0
...
2
votes
1answer
33 views
Proving a formula for $\int_{x=0}^\infty \frac{\sin(ax)x}{(x^2+1)^c} dx$ involving Gamma and Bessel K functions
In Mathematica,
$$\int_{0}^\infty \frac{\sin(ax)x}{(x^2+1)^c} dx
=\frac{2^{\frac{1}{2}-c}a^{-\frac{1}{2}+c}\pi^{\frac{1}{2}}\operatorname{BesselK}[-\frac{3}{2}+c,a])}{\Gamma[c]} ,$$
where a is a ...
1
vote
0answers
28 views
Explaining this estimation of $\pi$ with an integral
[This is potentially a duplicate] I was testing out integrals and I think I've found a way to evaluate a close approximation to $\pi$.
$$\int_{0}^{1} \dfrac{x^{4n}(1-x)^{4n}}{4^{n-1}(x^2 + 1)}\,dx$$
...
0
votes
0answers
23 views
Solving for a definite integral using a functional relation
Let $f$ satisfy the functional equation $$x = f(x)e^{f(x)}$$Then find the value of $$\int_{0}^{e}f(x)dx$$
My attempt:
$$\int_{0}^{e}f(x)dx$$
Using by - parts,
$$=[xf(x)]_{0}^{e}-\int_{0}^{e}xf'(x)dx$$...
1
vote
2answers
41 views
$\int_0 ^ \frac{\pi}{2}[\sin 2x (1+\cos 3x) ]dx$ . Here $[t]$ denotes the greatest integer less than or equal to $t$.
$\int_0 ^ \frac{\pi}{2}[\sin 2x (1+\cos 3x) ]dx$ . Here $[t]$ denotes the greatest integer function.
Can anyone give me a hint?
1
vote
1answer
45 views
Is there a way to find value for this integral?
I am trying to solve this integral by hand and here are my steps for doing that.
The integral is:
$$A=\int_0^{2 \pi} \frac{\cos(\theta)}{\sqrt{1 - a \cos(\theta)}} d\theta$$
Using the trigonometry ...
2
votes
2answers
72 views
How can I evaluate $\int _0^1\frac{\tan ^{-1}\left(3\sqrt{\frac{a}{4-a}}\right)}{\sqrt{a}\sqrt{4-a}}\:\mathrm{d}a$
I have the integral:
$$\int _0^1\frac{\tan ^{-1}\left(3\sqrt{\frac{a}{4-a}}\right)}{\sqrt{a}\sqrt{4-a}}\:\mathrm{d}a.$$
If I use $u=3\sqrt{\frac{a}{4-a}}$, I get
$$6\int _0^{\sqrt{3}}\:\frac{\tan ^{-1}...
0
votes
2answers
60 views
Evaluating $\int_{-a}^a x^{2n+1}\mathrm{d}x$ for all non-negative integers $n$ simultaneously
My assumption would be
$$\int_{-a}^a x\ dx=0$$
Am I on the right track here? Also, for indefinite integrals
$$\int (f)x\ dx$$
would this be correct as well?
Background
My professor raised this ...
1
vote
2answers
40 views
If $\int_{-1}^1 fg = 0$ for all even functions $f$, is $g$ necessarily odd?
Suppose for a fixed continuous function $g$, all even continuous real-valued functions $f$ satisfy $\int_{-1}^1 fg = 0$, is it true that $g$ is odd on $[-1,1]$?
My intuition is telling me that this is ...
1
vote
1answer
83 views
Solve the integral equation $\int_0^{-x}f(x')dx'= f(x) + x$.
Like it says, I'm playing around with even and odd functions and require a function such that $$\int_0^{-x}f(x')dx'= f(x) + x\,.$$ I can't think how to go about it, any help appreciated.
1
vote
0answers
36 views
weak continuity
Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n$, $u \in L^\infty\!\left(\left[0,T\right],H^1_0\!\left(\Omega\right)\right)$ and $u_t \in L^\infty \!\left(\left[0,T\right],L^2\!\left(\Omega\...
1
vote
0answers
25 views
Why does this improper integral converge
I can't understand a part of a proof. Here is the strange part isolated.
Let $1<q<2$. Then
$(\int_{|z-\tau|\leq R}\frac{d\tau}{|z-\tau|^q})^q = (\frac{2\pi R^{2-q}}{2-q})^q$
The integral is over ...
1
vote
0answers
33 views
Verifying the integral of ${\int\limits_0^{2\pi } {{e^{a\cos (\theta - b)}}f\left( {r,\theta } \right)} d\theta }$?
I tried doing the above integral in the following way. I am not sure if this is correct. Did I break any fundamental rule? Thank you.
Let, $f(r,\theta)=re^{j\theta}$.
$\begin{array}{*{20}{l}}
{\int\...
1
vote
1answer
25 views
Differential equation with non-elementary function
Can a solution be found to the following differential equation:
$$ \dfrac {dy}{dx}=e^{-x^2}$$ given the initial conditions $y(0) = c$ - some constant?
I know that $y(x)$ is a non-elementary function ...
2
votes
4answers
68 views
Determine $\int_{-\infty}^\infty e^{ipx - qx^2} dx$.
I have to evaluate the following integral:
$$\int_{-\infty}^\infty e^{ipx - qx^2} dx\,,$$
where $p \in \mathbb{R}$ and $q > 0$. I am suppose to use contour integration, but I am not sure what the ...
0
votes
0answers
26 views
Why does this simple equation involving an integral hold?
Within an exercise, I was wondering about this step of solving an equation:
\begin{align}
\int_0^s \frac{\dot{z}(t)}{f(z(t))} \text{d}t = s \\
\Leftrightarrow \int_{z(0)}^{z(s)} \frac{1}{f(z(t))} \...
2
votes
0answers
40 views
Can this integral be expanded as a power series?
I am looking at this integral:
$$I(x_1^2, x_2^2) = \frac{1}{4}x_1^2 x_2^2 \int_{-\infty}^{\infty} d\tau_3 \int_{-\infty}^{\tau_3} d\tau_4 \int_{-\infty}^{\tau_4} d\tau_5 \int_{-\infty}^{\tau_5} d\...
1
vote
1answer
59 views
What does this integral notation mean?
I'm talking about the integral part that is highlighted:
Should I interpret the top one highlighted as the upper bound of integration and the bottom one as the lower bound? That's the only ...
0
votes
0answers
25 views
Minimum of product of integrals
This is really a problem from Lagrangian mechanics.
Given an action integral:
S = $\int L_{t}(t,\dot{x}, x_{t})\cdot L_{\tau}(\tau , x', x_{\tau} )dt \mathrm{d} \tau$
I'm trying to find the path ...
0
votes
1answer
35 views
Understanding the substitution theorem of Riemann integration.
Let us say $f$ is an integrable function on $[a,b]$ and we want to evaluate $\int_a^b f(x)dx$ but often the calculation is not easy.So,we have a method of substitution.We substitute $x=\phi(t)$ where $...
4
votes
1answer
123 views
Evaluate $\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm d}x $.
Problem
Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm
d}x .$
Someone writes as follows
\begin{align*}
&\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 ...
-1
votes
0answers
23 views
Calculation partial derivatives
Lets assume
$Y = X^2$
$Z = \frac{1}{n} \sum_1^n Y_i$
$\frac{\partial Z}{\partial X} = \frac{X}{2}$
How partial derivative of $Z$ with respect to $X$ is Calculated?
More Detailed info here
2
votes
1answer
34 views
Is this a sufficient condition for the integral to be greater than $m$?
It is given that $f:\mathbb R \to \mathbb R$ is positive continuous function.
If it is positive then it means that it never goes below the $x-$ axis. I want to find a sufficient condition such that we ...
2
votes
0answers
16 views
Two integrals in d-dimensional spherical coordinates depending only on the relative angle
I have the following problem: I have two d-dimensional integrals, where the integrand has the following dependence:
$$\int\mathrm{d}^{d}x\mathrm{d}^{d}y\,f(\vert\vec{x}\vert,\vert\vec{y}\vert,\vert\...
0
votes
1answer
55 views
Is $n=2 $ the only solution of the below identity?
It is known that $\displaystyle\int_{-\infty}^{\infty} \frac{\sin \left( x\right )}{x} \mathrm{d}x = \int_{-\infty}^{\infty} \frac{\sin ^ 2\left( x\right )}{x^2} \mathrm{d}x$, I'm interested to know ...
0
votes
0answers
53 views
How to prove $v(\Omega)=\frac{1}{3}\iint_S\textbf{r} \cdot d\textbf{S} $?
Suppose the boundary of $\Omega\subset \mathbb{R^3}$ is a smooth surface $S$, prove $$v(\Omega)=\frac{1}{3}\iint _S\textbf{r} \cdot d\textbf{S}. $$
When $S$ under the general sphere coordinates $(x,y,...
-1
votes
1answer
40 views
Integral of Error Function [closed]
Is there a way to approximate this integral with a constant expressed in terms of $\delta$
$$\int_{0}^{1} e^{-\left(\frac{x^2}{2\delta^2 }\right)} dx$$
Thanks
-3
votes
1answer
31 views
Suppose $C$ is a piecewise smooth curve, how to prove this? [closed]
Suppose $C$ is a piecewise smooth curve, $f,g,h,w :C\rightarrow \mathbb{R}$ are continuous. Let $M=\max\sqrt{f^2+g^2+h^2}$ on $C$. Prove $|\int_C wfdx+wgdy+whdz|\leq M\int_C|w|ds$.
-5
votes
0answers
33 views
Will ∫_a^b▒〖f(x)□(24&dx*)〗 ∫_c^d▒〖g(y)□(24&dy)〗=∬_ac^bd▒〖f(x)*g(y)dy dx〗work for any f(x) and g(y)? [closed]
if ∫_a^b▒〖f(x)□(24&dx)〗 is convergent and ∫_c^d▒〖g(y)□(24&dy)〗is divergent then will the order of dy.dx change as dx.dy?
0
votes
2answers
33 views
Simplifying integrals with greatest integer in arguments
Suppose $ \int_{0}^{1}\sin( (x-[x]) \pi) dx$
This inside argument of this $\sin(x)$ spans $0$ to $\pi$, therefore I thought this integral would be $\int_{0}^{\pi} \sin(x) dx$
But it is infact, $ \int_{...
0
votes
1answer
26 views
The Integrability of the Nth composite for the Riemann integrable function
Generally we can't say " if both $g$ and $f$ are Riemann integrable on $\mathbb{R}$ then
the $g \circ f$ is Riemann integrable."
So more considering the specific case, Is the $f^n(x) (=f \...
2
votes
0answers
29 views
limit of integral with doubly period $1$ continuous function.
This is a problem that I recently saw in some lecture notes is proving a little more challenging that what I anticipated:
Suppose $f$ is a continuous function on $\mathbb{R}^2$ and such that
$$f(x+1,y)...
0
votes
1answer
54 views
Evaluating $\int _0^{\infty }e^{x\left(t-p\right)}p^{1-e^{-px}}\:dx$
How do I proceed with this integration? When I try integration by parts, the same thing keeps coming over and over again.
$$I=\int_0^\infty e^{x(t-p)}p^{1-e^{-px}}\,dx$$
4
votes
2answers
130 views
How can I solve this $\int\dfrac{1}{\sqrt{5e^{-2x}+4e^{-x}+1} } \mathop{dx}=?$
How can I solve this $$\int\dfrac{1}{\sqrt{5e^{-2x}+4e^{-x}+1} } \mathop{dx}=?$$
My attempt:
I substituted $e^{-x}=t$, $-e^{-x}\ dx=dt$, $dx=-\dfrac{dt}{t}$
$$\int\dfrac{1}{\sqrt{5t^2+4t+1 } }\left(-\...
2
votes
3answers
124 views
Verify “90% of scientists are alive today” with calculus
I am reading Richard Hamming's book and came across a back of the envelope calculation I am having trouble reconciling. The relevant assumptions in the calculation are:
assume the number of ...
0
votes
0answers
22 views
Max distance a motorcycle can travel? [closed]
This motorcycle goes on with two tires and have one extra with it. If the max distance a tire can go is 5 kilometers then what is the max distance the motorcycle can go with 3 tires and unlimited ...
1
vote
0answers
65 views
When is it possible to use complex analysis for solve the integrals?
"Is there a criterion, a clue that makes me think that certain integrals can also be solved through complex analysis and how to solve them?"
When I can't solve an integral, I use the ...
3
votes
0answers
84 views
Solving $\int \sqrt {f(x)} \ dx=\sqrt{\int f(x)\ dx}$
I am trying to solve the following differential equation:
$$\int \sqrt {f(x)} \ dx=\sqrt{\int f(x)\ dx}$$
My approach:
I started by differentiating both sides and got:
$$\sqrt {f(x)}=f(x) \frac{1}{...
-3
votes
2answers
53 views
If $f$ and $g$ are zero mean, then $f\cdot g$ have zero mean? [closed]
Let $f,g: \mathbb{R} \longrightarrow \mathbb{R}$ be differentiables and periodics functions, with minimal period $L>0$. If
$$\int_{0}^{L}f(x)=0 \quad \text{and} \quad \int_{0}^{L}g(x)=0$$
then
$$\...
0
votes
2answers
33 views
Evaluate the complex line integral $\int_\gamma\frac{z^5}{z^7+3z-10}\,dz$, $\gamma$ is the boundary of $D(0,2)$ oriented counterclockwise
This is a UW Madison analysis qualifying exam problem.
I think if $z\in\mathbb{C}$, $|z|\geq 2$ then $|z^7+3z-10|\geq|z|^7-3|z|-10>0$, so we can take $\gamma$ to be the boundary of $D(0,R)$ for any ...
1
vote
3answers
70 views
Integral of $\sqrt{1-\|x\|^2}$
I am trying to calculate the next integral:
$$\int_{Q}\sqrt{1-\|x\|^2}dx$$
where $Q =\{x\in\mathbb{R}^n: \|x\|\leq 1\}$ and $\|x\|$ is the usual norm of $\mathbb{R}^n.$
For the cases $n = 2$ and $n = ...
2
votes
1answer
48 views
Integral with delta function
In an exercise, I found after some time the following integral:
$$\int_{-\infty}^\infty \mathrm{d}x\,\mathrm{d}y\,f(x,y)\int_{-1}^1 \mathrm{d}z\,\delta\left(z-\frac{2-2x-2y+xy}{xy}\right)$$
In the end ...
