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.

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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' ...
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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 $J_{n+1} = \frac{3n-1}{3n} J_n$?
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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 ...
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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.
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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{\...
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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,\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\pi$. $$\int_{0}^{1} \dfrac{x^{4n}(1-x)^{4n}}{4^{n-1}(x^2 + 1)}\,dx$$ ...
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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, $$=[xf(x)]_{0}^{e}-\int_{0}^{e}xf'(x)dx$$...
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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?
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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 ...
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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}...
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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 ...
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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 ...
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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.
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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\...
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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 ...
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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\...
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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 ...
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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 ...
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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))} \...
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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\...
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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 ...
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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 ...
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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 $...
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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 ...
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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
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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 ...
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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\...
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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 ...
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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,...
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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
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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 $|\int_C wfdx+wgdy+whdz|\leq M\int_C|w|ds$.
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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?
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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_{...
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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 \...
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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)...
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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. $$I=\int_0^\infty e^{x(t-p)}p^{1-e^{-px}}\,dx$$
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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(-\...
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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 ...
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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 ...
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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 ...
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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}{...
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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 $$\...
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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 ...
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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
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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 ...

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