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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1answer
31 views

Book to learn Integration

I have studied real analysis from Baby Rudin and Bartle. I find a lot of difficulty in tackling integration problems, especially when integration is mixed with sequences and series of functions, and ...
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1answer
77 views

Two cool sums: Compute $\sum_{n=1}^\infty (-1)^{n-1}\frac{H_{2n+1}}{(2n+1)^3}$ and $\sum_{n=1}^\infty (-1)^{n-1}\frac{H_{2n+1}^{(2)}}{(2n+1)^2}$

How to prove $$S_1=\sum_{n=1}^\infty (-1)^{n-1}\frac{H_{2n+1}}{(2n+1)^3}=1+\frac{35}{128}\pi\zeta(3)+\frac{1}{48}\zeta(4)-\frac1{384}\psi^{(3)}\left(\frac14\right)$$ $$S_2=\sum_{n=1}^\infty (-...
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2answers
83 views

Solving $ \int \sqrt{1 + \tan(x)}\:dx$

I know this is lazy, but I was really hoping that someone could read over my work on this integral and let me know whether I've made any errors. Here I will address the integral: \begin{equation} ...
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0answers
11 views

solve the initial value problem by the laplace transform. [on hold]

solve the initial value problem by the laplace transform.show all details. y''+2y'+5y=50t, y(0)=-4, y'(0)=14
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0answers
35 views

Verify the Solution of an Integration

I have following interrogation to be solved: $$I=\int_{0}^{\infty}\frac{\Gamma \left(a,b \sqrt{x}\right)}{x+1}\,dx; a,b>0$$ Since this is a complicated one I first try in Mathematica which gives ...
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0answers
16 views

How to take integral of $\frac{pdf}{cdf}$ of two normal distributions with different means?

Is it possible at all to find the closed-form solution? If $x=0$ it's obvious. But what to do for $x\neq 0$? $\int\limits_y^{\infty } \frac{e^{-\frac{(\xi -x)^2}{2 \sigma ^2}}}{\sigma \left(1+ \text{...
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0answers
54 views

Variations Calculus. How to compute some variations?

I have the following situation: I'm trying to prove last point: the metric is invariant to re-parameterization of $f$. So, the left member can be rewritten in the following way: $$a^2 \int^{1}_{0}...
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1answer
54 views

How to evaluate the following derivative of integral function [on hold]

$$\frac{d}{dt}\left(\int_{0}^{t}\frac{1 - e^{a(t-x)}\operatorname{erfc}\left(\sqrt{a(t-x)}\right)}{\sqrt{x}(x+b)}\,dx\right)$$
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1answer
60 views

How are derivatives and integrals exempt from the chain rule? [on hold]

A question I never considered when studying calculus is why the chain rule seems to fail when applied to derivatives and indefinite integrals. For example, according to the chain rule, the derivative ...
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3answers
60 views

Solving integral of $\ln(A)/\sqrt{A}$

I am working on a physics problem that requires solving the following integral: $$\int \frac{\ln(r^2+R^2+2rRu)\,du}{(r^2+R^2+2rRu)^{\frac{1}{2}}}$$ However, I don't know how to approach it, as it is ...
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0answers
38 views

Why does integration have to be the reverse operation of differentiation? [duplicate]

I am in High school and when I was learning calculus, we were taught that integration is nothing but the reverse of differentiation. But I really don't get it why does that needs to be the case. ...
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2answers
53 views

Helpful Hints to solve a difficult Integral

I am attempting to solve this integral/problem I found on brilliant. Given$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x^{2}\cos(x)}{1+\exp(x^{2}\sin(x))}\,dx $$ converges to $\dfrac{\pi^{a}-b}{c}$, ...
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1answer
15 views

Integral convergence using Taylor expansion [on hold]

How can I analyze for which $s$ the integral $\int_{0}^{1} \frac{x^s} {\sqrt[3] {1+x^7} - 1} dx$ converges? I simplified $\sqrt[3] {1+x^7}$ using the binomial formula but I don't know how to continue.
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1answer
103 views

Evaluate $\int_{0}^{\frac{\pi}{4}}\tan xdx $ using idea of Riemann Sum

Evaluate $$\int_{0}^{\frac{\pi}{4}}\tan x\,dx$$ using Riemann Sum. My Attempt: $$\int_{0}^{\frac{\pi}{4}}\tan x\, dx=\frac{\pi}{4}\int_{0}^1\tan\left(\frac{\pi}{4}x\right)dx=\frac{\pi}{4}\lim_{n\to\...
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5answers
75 views

Solving $\int \frac{e^{2x}}{1+e^x} \, dx $

Problem: $\int \frac{e^{2x}}{1+e^x} \, dx $ My book says to divide in order to solve by getting $\int e^x-\frac{e^{x}}{1+e^x} \, dx $ but how am I supposed to divide? I tried long division but ...
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2answers
62 views

Integral Table Result

To prove the result $$\int_{1}^{e}\frac{\ln(x)}{(\alpha +\ln(x))^{(\alpha+1)}}\,dx = \frac{e}{(\alpha+1)^\alpha} - \frac{1}{\alpha^\alpha}$$ I have used the substitution $t = \alpha +\ln(x)$ which ...
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0answers
23 views

Inequality: norm of differential forms and mass of singular simplex

Let M be a Riemannian manifold of dimension $n$ with a differentiable $p$-form $\omega\in \Omega^p(M)$. Let $\sigma\colon \Delta^p\to M$ be a differentiable singular $p$-simplex. We may assume that $\...
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0answers
22 views

Product of Bessel and Hankel functions integral identity

The following identity holds in the case where the upper limit of integration is infinite: $$\int_{0}^{\infty} dz\, z^{\mu+1}H^{(1)}_\nu(az)H^{(2)}_\nu(bz)J_{\mu}(cz) \ = \ \sqrt{2}c^\mu\frac{\Gamma(\...
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1answer
73 views

An integral for the difference of zeta functions $\zeta (s-1)-\zeta(s)$

Starting with: $$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x$$ How can we prove for $s > 2$ the following conjecture: $$\zeta (s-1)-\zeta(s)={\...
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1answer
41 views

Integrals Over Complex Domains

In Complex Analysis texts, one often sees integrals of functions $f : A \subset \mathbb{C} \to \mathbb{C}$ along contours $\gamma : [0,1] \to \mathbb{C}$. But I've never seen discussion of integrals ...
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1answer
33 views

Is this Change of Variable Theorem fine as it is, or should we specify the domain that $f$ is continuous on?

One of my instructors provided me with the following Change of Variable Theorem: Let $a < b$. Let $f$ be a continuous function. Let $g$ be a function with a continuous derivative on $...
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2answers
59 views

Can this be done without integration by substitution?

Assuming that g is a continous function in the interval $[a,b]$ and $c$ is a non-zero constant, I have to show that: $\int^b_ag(x)dx=\int^{b+c}_{a+c}g(x-c)dx$ Now my question is whether this is ...
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0answers
25 views

Where to start with an intergral involving two Heaviside step functions?

This question is a follow-up to a previous question. I am dealing with the following type of integral: $$ f(\vec x) = \int_{\mathbb R} \text{d}^3y\; (\vec x\cdot \vec y)^n H(A-|\vec y-\tfrac{1}{2}\...
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3answers
388 views

Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$

How to show that the integral $$\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx$$ equals to $\frac{3 \pi ^2}{160}$? I've already verified this numerically ...
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0answers
38 views

How do I work out the distance of Venus orbit mathematically? [duplicate]

How do I work out the distance Venus travels in one orbit mathematically? I know the parametric equations for its ellipse but I need to work out the total distance travelled in its orbital period.
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1answer
35 views

The proof for the Lebesgue differentiation theorem

Im looking at the proof of the Lebesgue differentiation theorem on wikipedia: https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem#Proof I don't see why this line is true. This looks like ...
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2answers
94 views

Is $-\int_{0}^{\infty} \bigg( \exp(-\dfrac{\pi}{4}x²)) \bigg) \;dx+\bigg(\sum_{n=0}^{\infty }\bigg( \exp(-\dfrac{\pi}{4}n²)\bigg)=\dfrac12$ true?

The following sum may it is easy for computation $$-\int_{0}^{\infty} \bigg( \exp(-\dfrac{\pi}{4}x²)) \bigg) \;dx+\bigg(\sum_{n=0}^{\infty }\bigg( \exp(-\dfrac{\pi}{4}n²)\bigg)=\dfrac12$$ The sum ...
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1answer
47 views

How to change limit of this integral

Let $ I$ = $$\int_0^{a}\int_0^{a-x}\left(f(x,y)\right)dydx$$ then I need to find the new integral by changing the variables $x+y = u$ and $xy =v$, Now for u I can easily find the limits as they vary ...
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1answer
60 views

Evaluating $\int_{0}^1\left(\int_0^{1}fdx\right)dy$, where $f(x,y)=\frac12$ for $x$ rational, and $f(x,y)=y$ for $x$ irrational

So, while solving problems on double integral I came across this weird problem: Let $$f(x,y) = 1/2,\ \forall x\in \mathbb Q$$ $$f(x,y) = \ y, \ \forall x\in \mathbb Q^c$$ then find $$\int_{...
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1answer
63 views

Is this a correct usage of triple integrals?

The problem states: Find the volume of a region bounded by $x^2+y^2=4, \quad y=z, \quad y+z=4 $ So the idea is to triple integrate over the given region. Since sketching stuff in 3D is rather ...
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5answers
64 views

Partial fractions ODE first order

Given the problem: $\frac{dx}{dt} = 3x(x-5)$ The answer is supposedly: $\dfrac{40}{8-3e^{-15t}}$ with $x(0) = 8$ first off you can't have a negative power in the denominator it should just be $e^{...
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0answers
65 views

Is my understanding of differential forms integration correct?

I'm trying to understand the definition of the integral of a general $k$-form $\omega$ over a parameterised $k$-manifold $M$ :$$\intop_{M}\omega=\intop_{D}\omega\left(\frac{\partial\mathbf{X}}{\...
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1answer
44 views

Trapezium rule vs integration

Wikipedia says the trapezium rule is "a technique for approximating the definite integral" (my emphasis). Isn't the trapezium rule identical with definite integration as the number of strips gets ...
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0answers
34 views

Modelling the shape, volume, and the surface area of a shampoo bottle.

I want to model the volume and the surface area of the object that can be seen in the images below. Front view of Bottle 1 Side view of Bottle 1 Front view of Bottle 2 Side view of Bottle 2 I ...
2
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1answer
60 views

An integral inequality about $f(f(x))$

Let $f:[0,1]\rightarrow \mathbb{R}$ be continuous and monotonically increasing. Assume $f(0)=0$,$f(1)=1$. Prove:$$\int_0^1{f\left( f\left( x \right) \right) dx}\le 2\int_0^1{f\left( x \right)}dx $$...
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0answers
34 views

Integration of periodic function

Let $a$ a T-periodic function. It is obvious that we have $$n\int\limits_0^T a (s) = \sum\limits_{i = 0}^{n - 1} {\int\limits_{iT}^{(i + 1)T} a (s)ds} $$ I want to estimate the following quantity$$\...
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2answers
35 views

Is this correct way of solving this taks that relates to integral calculus

The task says find $G'(x)$ ($G$ differentiated) if $G(x) = \int_{2x}^1f(t)\,dt.$ I have found it like this: $$G(x) = \int_{2x}^1f(t)\,dt = G(x) = F(t)|_{2x}^1 = F(1) - F(2x) / \frac{d}{dt} = 0 -F^,(...
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0answers
57 views

Finding parameters for function to be integrable

Here's the function: $h(x) = \begin{cases} \cot x & \text{if $x \in [-\pi/2,0)-\{-\pi/3\}$}\\ p & \text{if $x=-\pi/3$}\\ q & \text{if $x=0$} \end{cases}$ Find the value of ...
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3answers
83 views

What does it mean to integrate a complex function over a real domain?

In complex form, we know that the $n$-th Fourier coefficient of a function $f$ is given by $$\int_{-\pi}^{\pi} f(\theta)e^{-in\theta} d\theta.$$ My question: What exactly does it mean to integrate ...
3
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1answer
57 views

$f(x) = \int_{k=0}^x e^{f(k)}\, dk$

I have the function $$f(x) = \int\limits_{k=0}^x e^{f(k)} dk$$ How can I solve for an explicit formula of $f$? In my current solution, it becomes undefined past a certain value of $x$ Atempts: ...
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1answer
47 views

An application of Green theorem

Greens theorem states: $$ \oint_C P dx + Q dy = \int \int_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dxdy $$ I have to calculate: $\oint_γ x y^2 dx + (x+y) dy$ in the domain: ...
13
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4answers
306 views

Integration with $\ln(x)$ in the denominator

Find$$\displaystyle\int_1^\infty\frac{(x^2-1)(x^4-1)(x^6-1)}{\ln(x)(x^{14}-1)} dx$$ I tried simplifying the terms without logarithm $x^2-1=(x-1)(x+1)\\x^{14}-1=(x^7-1)(x^7+1)$ to see if any ...
5
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1answer
48 views

Conditional and absolute convergence of an integral

If you know that $$\int\limits_x^{+\infty} e^{-t^2}\mathrm dt=e^{-x^2}\biggl(\frac{1}{2x}+o\biggl(\frac{1}{x^2}\biggl)\biggl)$$ examine conditional and absolute convergence of this integral: $$\int\...
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0answers
29 views

Integrating the product of two normal pdfs

I have been working on some problems on Gaussian processes and came across the formula $$ \int \mathcal{N}\left(a ; Bc, D\right)\mathcal{N}\left(c; e, F\right) dc = \mathcal{N} \left(a; Be, D+BFB^...
2
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1answer
91 views

How to Evaluate proper Integral.

Recently I stumbled upon an integral and its solution in a physics article but I couldn't understand how it was evaluated.I have plotted the function and it indicates that the value of the integral ...
3
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2answers
142 views

Integral $\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}dx$

How do I evaluate $$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$ To my knowledge the following integral should be related to the Gamma function. I ...
5
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1answer
67 views

Need help evaluating this improper double integral

Need help evaluating this improper integral $$\iint_D (1-x+y)e^{(-2x+y)}\,dx\,dy$$ where $D$ is the region: $0 \le y \le x $. I've tried doing the substitution: $u = -2x +y \\ v = 1-x+y$ which ...
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0answers
34 views

Find $\int (z^4+4z)e^z \cos^2 z \,\mathrm dz$ [closed]

$$\int (z^4+4z)e^z \cos^2 z \,\mathrm{d}z$$ To find this complex integration over the circle $|z-2|=5$
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0answers
53 views

How can I calculate the surface area of an umbrella? [closed]

Umbrella shape I want to find the surface area of a beach umbrella, similar to what is shown in the image below. I recognize that it is made by intersections of half-cylinders. however, I still can't ...
0
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2answers
43 views

What would be the step by step solution of this double integral by changing it to Polar coordinates?

$$\int_0 ^ {1} \int_{-\sqrt{x-x^2}} ^ {\sqrt {x-x^2}} (x^2+y^2) ~dy~dx$$ My findings are: $$\int_?^? \int_?^? r^3 ~dr~d\theta$$ Region is the circle of radius $~\frac{1}{2}~$ centered at $~(\frac{...