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.
70,221
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If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
Hint is to use integration by part but I can't find a good reference for ...
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2
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56
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I need help with integrating an expression with multiple $x$ and $y$ terms [closed]
The equation is $x^2+y^2 = 3\sqrt{2} x - 5\sqrt{2} y +2xy$. I need to try and find the area under the curve and above the x axis between $x = 0$ and $x = 3\sqrt{2}$. I've heard that implicit ...
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2
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52
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Defining formulas for first-order linear differential equations.
When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And ...
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3
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50
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Is this weird function with argument in the integrand continuous? (Fundamental Theorem of Calculus)
Let $f:I \rightarrow \mathbb{R}$ with $I$ interval and $f \in C^\infty(I)$. If $t_0 \in I$, we know by the FTC that $F:I \rightarrow \mathbb{R}$ given by $F(x)=\int_{t_0}^x f(t)dt$ is continuous. But ...
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2
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69
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Solve an integral by bringing quadratic trinomial to canonical form
The problem is the following:
$$\int{\sqrt{4x^2+x}dx}$$
Now once gotten to canonical form of a quadratic trinomial, $ax^2+bx+c=a(x-(\frac{-b}{2x}))^2-\frac{b^2-4ac}{4a}$, the intregral looks like this:...
- 143
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31
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visualization of triple integrals with iterated bounds in different coordinate systems
Lately i've been trying rather hard to write a code in cpp or python that not only calculates the result of a given triple integral, but also depicts the shape of the volume defined by the integral. ...
2
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2
answers
188
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Find $\int\frac{1}{2\sin (x)+3\cos (x)+1}$ $\Tiny{dx}$
Question
Evaluate the following integral: $\int\frac{1}{2\sin (x)+3\cos (x)+1} \small{dx}$
Now, I've tried a couple of different substitutions and integrating partially but unfortunately, to no ...
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29
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Integration by Parts Table Technique With Exponential and Polynomial Higher Order
I come to you with a rather simple question in need of a reference or two from more knowledgable sources. Here was the simple integration by parts problem:
However, a rather peculiar table (book-...
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Need help solving a wind power problem involving wind speed, energy production, and optimization
I am working on a wind power problem and need some help. Here is the problem:
During a windy day, the wind speed at a wind turbine can be described by the model:
$$
v(x)= 11 \sin(0.11x - 0.89) + 28, \...
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2
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Help needed to calculate the derivative of $\int_0^t \cos(x^2) d x$
I'm currently working on a calculus problem and I could use some help. The question is to find the value of $F^{\prime}(\sqrt{\pi})$ if $F(t)=\int_0^t \cos \left(x^2\right) d x$.
I'm not sure how to ...
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What techniques can I use to solve $ \int$ $ \small{\dfrac{2}{t^2(a-b)+2ct+(a+b)} \space dt}$?
I am trying to calculate the following integral:
$$\int \dfrac{2}{t^2(a-b)+2ct+(a+b)}dt$$
I've been going through an old university textbook (at the age of $44$) just to see if I fully understand ...
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Integrating $y' = \sin^2 y$
I'm trying to solve: $$y' = \sin^2 y$$
It is a separable variable differential equation, so I arrive to
$$\int\frac{dy}{\sin^2 y}=\int dx$$
I use the identity $$\sin^2 y=\frac{1-\cos(2y)}{1}$$ and ...
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49
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Difference between $\int$ and $\int_{-\infty}^{\infty}$ [duplicate]
I was browsing questions and I found this one which uses a definite integral from negative infinity to positive infinity. I then wondered if there was a difference between that and the indefinite ...
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Rephrasing Double Integral Equation
Suppose we have a equation $$\iint_{-\infty }^{\infty} xy \frac{1}{2\pi \sigma^2}\exp\left(-\frac{(x-\mu_x)^2+(y-\mu_y)^2}{2\sigma^2}\right) dx dy$$
What property of the integral has been used to ...
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1
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80
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Find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$.
How do I find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$?
I know that the complex Fourier transform is $\hat{f}(k) = \displaystyle \int_{-\infty}^{\infty} f(x)e^{-ikx}\ dx$.
...
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1
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Show the existence of a limit of an integral
I was presented the following problem at Calculus class:
Let $f: (0, \infty) \to \mathbb{R}$. We know that $f$ is Riemann integrable in every compact interval of $(0, \infty)$, and that the limit $\...
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Integration signum of a continuous function
Let $f:[a,b]\longrightarrow \mathbb{R}$ be continuous, and denote by $\mathrm{sgm}(x)$ the signum of $x$. Define $g(x):=\mathrm{sgm}(f(x))$ for all $x\in [a,b]$. We assume we don't know the expression ...
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Double Integral Split Rule
Suppose we have a function:
$\iint_{-\infty }^{\infty} (ax+by)f_{xy}(x,y) dx dy$
I would like to know what property of the integral rule has been used to rephrase the above equation into like below:
$...
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Find the volume below $z = \sqrt{1 - r^{2}}$ and above the top half of the cone $z = r$
I have made an attempt at this question:
Find the volume below $z = \sqrt{1 - r^{2}}$ and above the top half of the cone $z = r$
I solved for $r$ first, then I calculated:
\begin{equation}
\int_{0}^{...
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An integral problem that has been bothering me for a year
This is the triple integral, the lambda here is going to 0+, the integration result is as follows, I hope someone can help me to answer, thank you very much，enter image description here
enter image ...
2
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1
answer
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How does log come into play?
$$\begin{align*}I & = 2\int_{0}^{\frac{1}{\sqrt{2}}} \dfrac{\sin^{-1} x}{x}\ dx - \int_{0}^{1} \dfrac{\tan^{-1} x}{x} \ dx \\ & = 2\int_{0}^{\frac{\pi}{4}} \dfrac{\theta \cos \theta}{\sin \...
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2
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Definite integral over an infinite product
Evaluate the following integral $$\int_0^\infty\frac{x+1}{x+2}\cdot\frac{x+3}{x+4}\cdot\frac{x+5}{x+6}\cdots dx$$
When I saw this, I was pretty sure that the infinite term must telescope or it must ...
1
vote
0
answers
21
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Centre of Mass in x,y plane
Given an area $R$ in x,y plane, and the density is $\rho(x,y)$ at $(x,y)$, then we can use double integrals to calculate the centre of mass coordinates. Now my question is, intuitively, given any non-...
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Why don't the bounds in this definite integral change?
The question
This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
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Limits of Riemann sums left and mid endpoint rule.
In my calculus class we have begun talking about integrals. In particular we have begun talking about Reimann sums and how through the limit of a Reimann sum we can integral. But so far all our ...
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4
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How do I solve $ \int_{0}^{1}\frac{x^2}{\sqrt{3+x^2}}dx$?
I had to solve this integral:
$$
\int_{0}^{1}\frac{x^2}{\sqrt{3+x^2}}\mathrm{d}x
$$
by the substitution $x=\sqrt{3}t$ the indefinite integral can be written as:
$$
\int \frac{x^2}{\sqrt{3+x^2}}\;\...
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Second Order Homogenous ODE solve with Real Analysis Integration topics [duplicate]
Question:
Suppose that $u\in C([a,b])$ is twice continuously differentiable, $V\in C([a,b])$, $V(x)\geq0$ for all $x \in [a,b]$ and
$$
-u''(x) + V(x)u(x)=0, \;\; x\in [a,b],
$$
$$
u(a)=u(b)=0
$$
Prove ...
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If $f$ is $L^1$ and continuous, then does $\int |f(a-x)-f(a)|dx<\infty$ hold?
Let $f:\mathbb R\to \mathbb C$ be a $L^1$ and continuous function, and $a\in\mathbb R.$
Then, does $$\int_{-\infty}^\infty |f(a-x)-f(a)|dx<\infty$$ hold ?
Now, I have $\int_{-\infty}^\infty |f(x)|...
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Stokes Theorem on manifolds with dense corners
I am currently working on a project, where I would (ideally) like to apply Stokes theorem on a Manifold with corners. I have found various sources, which justify this application. Except one thing:
In ...
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0
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Evaluate the following trigonometric integral with exponential function
Find the value of $$\int_0^\pi\mathrm{e}^{\mathrm{e}^{\cos\left(x\right)}}\cos\left(\sin\left(x\right)\right)\cos\left(\mathrm{e}^x\sin\left(\sin\left(x\right)\right)\right)dx$$
How to solve this ...
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1
answer
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Integration of Cahn-Hilliard-Oono equation
I am currently reading a paper on the Cahn-Hilliard-Oono equation with Neumann boundary conditions:
\begin{align}
\frac{\partial u}{\partial t} + \epsilon u &+ \Delta^2 u - \Delta f(u) = 0, \quad \...
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Is there a way to show multtiple integrals without using muliple integral symbols or the I symbol? Also can we say the -1 derivative is the integral?
How do I show multiple integrals (20th integral)?
Can I use the -1 derivative to show an integral?
Can I use the -1 integral ( if its symbol exists) to show a derivative?
3
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5
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Is there any other method to compute $\int_0^{\frac{\pi}{2}} \frac{x}{\sec x+\csc x} d x$?
Rationalization sometimes makes our life easier
Letting $x\mapsto \frac{\pi}{2}-x$ transforms the integral to
$\displaystyle I=\frac{\pi}{4} \int_0^{\frac{\pi}{2}} \frac{1}{\sec x+\csc x} d x=\frac{\...
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Integration a zero valued function product other function
Suppose I have a function $f(x) = 0$ for all $x$ except $0$. Now, $f(x) g(x)$ will also be zero for all $x$ except $0$ (or may be at $0$). Now, if try to find the value of $\int_{-\infty}^{\infty} f(x)...
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Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
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$\lim_{n\to\infty} \left(\frac{e}{n}\right)^n \int_0^n |x(x-1)(x-2)\dots(x-n)|dx$
What is
$$
\lim_{n\to\infty} \left(\frac{e}{n}\right)^n
\int_0^n \left| x(x-1)(x-2) \cdots (x-n) \right| \, dx?
$$
Context: I was trying to find an asymptotic expression for the total area of the ...
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change of variables formula in multiple integration
Use a change of variable in a double integral to compute the area of the square with vertices $(-3, 0)$, $(0, 3)$, $(3, 0)$, and $(0, -3)$. After the change of variable, the double integral should ...
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Are there arguments for not using Cauchy Principal Value when there's odd singularity?
Question: Are there arguments or examples that shows I should not always use Cauchy Principal Value when Riemann's integral is not defined for mathematical applications?
I take the case which $f(x)$ ...
- 1,136
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25
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Du Bois-Reymond criterion for Riemann-integrability
Function f \in \textbf{R} ([a,b]) (1) \Leftrightarrow f is bounded on [a,b] and for all ...
0
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1
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Integral Identities for $S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$ and $\zeta(x)$ \ $\zeta(y)$
Define $$S(x,y)=\sum_{k=1}^\infty\frac{y}{k^x(y+k)}$$ This is a generalization of the harmonic function ($n=2$). There are many ways we could relate this to $\zeta(x)$. For example $$\lim_{y\...
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What are the strategies to deal with intractable integrals encountered when solving ODE using variational method.?
I am trying to solve a nonlinear ODE: $$u_{xx}+\tan^2(x)u+gu^2=0$$ using the variational method, and I encountered an intractable integral. What are the strategies to deal with intractable integrals ...
2
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1
answer
38
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Ratio of an increasing monotonic function and its integral is infinity
Inspired by the fact that for functions at the form: $f(x) = \frac{1}{x^\alpha}$ where $\alpha \ge 1$, the intgeral: $\int_{0}^{1} f(x)dx$ diverges to $\infty$ , and the ratio: $\frac{f'(x)}{f(x)}$ ...
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1
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Uniform convergence and improper integral
I was thinking about this claim (Real analysis question)
let ${f_n}$ be a series of continuous functions that converges uniformly to $f(x)=0$ on the interval $[1,\infty)$ and that satisfies the ...
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1
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Calculate $ \int_{a}^{b} e^x \mathrm{d}x $with the aid of the Riemann subtotal.
Calculate $\int_{a}^{b}e^x \mathrm{d}x$ with the aid of the Riemann subtotal.
I know the Riemann subtotal and it is defined as follows: $S_n= \sum\limits_{i=1}^{n}{} f(ξ_i)\Delta x_i$. However, I have ...
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3
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1
answer
70
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Show $\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi$
I am tring to prove
$$
\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi.
$$
Numerical integration in Mathematica (truncating the integration bounds on $...
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0
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14
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Meaning of boundary = 0 for a singular k chain: Two interpretations
For a singular k-chain - denoted by $s^k$ - there is a corresponding definition of a boundary, that is:
$$\partial s^k = \sum_j l_j \partial c_j^k,$$
wherein $l_j$ is an integer coefficient and the $...
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votes
1
answer
59
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Is there any algebraic manipulation to evaluate $\int \big( f(x)+af'(x) \big) e^x \text{d}x$, where $a$ is a real parameter $\ne 1$?
I knew that
$$\int \big( f(x)+f'(x) \big) e^x \text{d}x = e^x f(x) + c$$
For instance,
$$\int \big( \sin(2x)+2\cos(2x) \big) e^x \text{d}x = e^x \sin(2x) + c$$
Also,
$$\int \big( \sin(x)\cos(2x)+\cos(...
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4
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212
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Prove that this expression is equal to $\pi$
Today, on the auspicious $\pi$ day, I saw on a local chat group $$\pi=4\phi^2\left(\phi^2+2\sqrt{\phi}\right)\left(\int_{\ 0}^{\ \infty\ }e^{x^2}\frac{\ \sin\left(x^2\right)\ }{x^2}dx\right)^2$$ I ...
1
vote
1
answer
61
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Double integral where integration limit of the inner integral is integration variable of outer integral
I am trying to replicate a paper. I want to prove that:
$$\sigma^{-1} \int_t^T k \int_s^T e^{-\rho (z-s)} x(z) dz ds = k \int_t^T m(z-t) x(z) dz$$
where $m(s) = (\sigma \rho)^{-1} (1-e^{-\rho s})$. In ...
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0
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27
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What's the meaning of the integral of the derivative of a distributin function $F$ if $F'(x)$ exists a.e.?
Lemma 2.2 on page 37 of Allan Gut's Probability: A Graduate Course (2nd edition) says :
Let F be a distribution function. Then:
(a) $F'(x)$ exists a.e., and is non-negative and finite.
(b) $\int_{a}^{...
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